On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity

University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange

Masters Theses Graduate School

8-2015

Josh Michael Lipsmeyer University of Tennessee - Knoxville, [email protected]

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Recommended Citation Lipsmeyer, Josh Michael, "On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity. " Master's Thesis, University of Tennessee, 2015. https://trace.tennessee.edu/utk_gradthes/3490

This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council:

I am submitting herewith a thesis written by Josh Michael Lipsmeyer entitled "On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Mathematics.

Alex Freire, Major Professor

We have read this thesis and recommend its acceptance:

Tadele Mengesha, Mike Frazier

Accepted for the Council: Carolyn R. Hodges

Vice Provost and Dean of the Graduate School

(Original signatures are on file with official studentecor r ds.) On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity

A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville

ii I would like to dedicate this work to my family: Mom, Dad, Joey, Brandi, Tim, and Shannon Your love, support, and encouragement has never waned for one second and for that I am truly thankful.

iii Acknowledgments

I would like to thank my advisor, Dr. Alex Freire, for his insights, guidance, and encouragement. I would like to also thank my committee members Dr. Mike Frazier and Dr. Tadele Mengesha for their contribution to my development in mathematics. I would also like to thank the geometry group at the University of Tennessee for always making themselves available to talk through problems and oﬀer their own valuable insights. Finally, I would like to thank Dr. Tom McMillan from the University of Arkansas at Little Rock for the pivotal role he played in laying the foundation of my career in mathematics, his constant encouragement, and friendship.

iv “Everybody can be great. Because anybody can serve. You don’t have to have a college degree to serve. You don’t have to make your subject and your verb agree to serve. You don’t have to know about Plato and Aristotle to serve. You don’t have to know Einstein’s theory of relativity to serve. You don’t have to know the second theory of thermodynamics to serve. You only need a heart full of grace. A soul generated by love.” - Martin Luther King, Jr.

“We all die. The goal isn’t to live forever, the goal is to create something that will.” - Chuck Palahniuk

v Abstract

The ”Fluid Ball Conjecture” states that a static stellar model in General Relativity is spherically symmetric. This conjecture has been the motivation of much work since first studied by Avez in 1964. There have been many partial results( ul-Alam, Lindblom, Beig and Simon, etc) which rely heavily on arguments using the Positive Mass Theorem and the equivalence of conformal flatness and spherical symmetry. The purpose of this thesis is to outline the general problem, analyze and compare the key differences in several of the partial results, and give existence and uniqueness proofs for a particular class of equations of state which represents the most recent progress towards a fully generalized solution.

vi Table of Contents

1 Introduction 1

2 Existence and Uniqueness of Static, Spherically Symmetric Solu- tions 5 2.1 Derivation of the System of Equations ...... 6 2.2 The Singular Point ...... 17 2.3 Extending the Solution Uniquely ...... 19

3 Uniqueness of the Static Stellar Model 23 3.1 Statement of the theorem...... 29 3.2 The “reference system” of O.D.E...... 31 3.3 Spinor approach to the Positive Mass Theorem...... 32 3.4 Strategy of proof of the Main Theorem...... 34 3.5 Auxiliary system of diﬀerential equations...... 38 3.6 The critical set and the oscillation set...... 41 3.7 The conformal factor on the regular set...... 46 3.7.1 Accumulation of discontinuities...... 49 3.8 Convergence of a sequence of conformal factors...... 51 3.9 Conclusion of the proof...... 54

4 Physical Constraints on Stellar Models 64 4.1 Buchdahl’s Inequality ...... 65

vii 4.2 Constraints on the Adiabatic Index ...... 70 4.3 Necessary and Suﬃcient Conditions for a Finite Radius ...... 76

Bibliography 81

Appendix 86

A Spin Geometry 87 A.1 Dirac Spinors ...... 87 A.2 Spin Manifold and the Diﬀerential Structure of the Dirac Bundle . . . 92 A.3 Dirac Operator and the Schr¨odinger-Lichnerowicz formula ...... 97 A.4 Witten’s expression for the mass ...... 102

Vita 108

viii Chapter 1

Introduction

The “Fluid Ball Conjecture” seems to have first been addressed by Avez in 1964.[23] The conjecture is concerned with equilibrium configurations for static stellar models. As in the Newtonian case [21], the belief is that a static stellar model equilibrium configuration always attains spherical symmetry. Partial results to the conjecture were attained in the 1970’s and 1980’s. Kunzle [20], Kunzle and Savage [15] made contributions to the conjecture but used very restrictive equations of state. ul-Alam [24],[25] brought in the use of the Posititve Mass Theorem [28]. ul-Alam’s work in the 1980’s unfortunately relied on unphysical equations of state to complete the proof. Lindblom [22] succeeded in proving spherical symmetry in the case of uniform density stars using ul-Alam’s work and the use of Robinson-type identities that had previously been used in proving uniqueness of static black holes.[27],[26]. A drawback to the work of ul-Alam and Lindblom in the 1980’s was that their method of proof required the existence of a “reference spherical stellar model” with the same mass

and surface potential Vs as their static stellar model. To complete the proof using their method they assumed the existence of the “reference model”, without proof. In 1994 joint work of ul-Alam and Lindblom [29] proved the existence of the ”reference spherical stellar model” under certain restrictions on the equation of state. This represented the most complete work up to that point. Given a static stellar model

1 and assuming an equation of state satisfying certain properties, the “reference stellar model” existed and the procedure developed in the 1980’s utilizing the Positive Mass Theorem showed that the static stellar model must in fact be spherical. In 2007 the most recent result pertaining to the “Fluid Ball Conjecture” by ul-Alam used a spinor norm weighted scalar curvature integral. It was this integral that had been used by Witten to prove the Positive Energy Theorem in n-dimensions[30].

The method of proving spherical symmetry using the Positive Mass Theorem has been the standard method of proof since it was first utilized by ul-Alam. The procedure is to start with a static stellar model with a certain given equation of state. The goal is to find a conformal factor so that the mass of the conformal metric is zero and the conformal scalar curvature is non-negative. The Positive Mass Theorem then implies that the conformal metric must in fact be flat. In the conformally flat case , Avez [23] showed that the original geometry had to be spherical. The difficulty in this method is showing the non-negativity of the conformal scalar curvature. The conformal factor is modeled after the conformal factor for the “reference spherical model”. This was the source of difficulty in ul-Alam and Lindblom’s work in the 1980’s. In order to show existence of the “reference stellar model” and the non- negativity of the conformal scalar curvature certain restrictions were placed on the equation of state. All of the modifications to this method revolved around restrictions on the equation of state.

Point-wise non-negativity of the conformal scalar curvature is a strict requirement. In an eﬀort to relax this condition, the use of a spinor norm weighted scalar curvature integral was utilized by ul-Alam in 2007. This allows the point-wise non-negativity to be relaxed as long as the overall negative contribution to the integral of the conformal scalar curvature is small. The scalar curvature integral is precisely the integral used by Witten in his proof of the Positive Energy Theorem. In ul-Alam’s work a conformal

2 factor is defined as a limit of conformal factors. In the limit, the scalar curvature integral with the scalar curvature in the original metric is shown to go to zero. The scalar curvature integral equaling zero implies the existence of a global convariantly constant spinor field. It is known that spinors are a type of “square root” of a vector so the global covariantly constant spinor field allows us to define a global covariantly constant frame field. This implies that the space is flat. Classical arguments [23] then imply that since the conformal geometry is flat, then the original geometry must be spherically symmetric.

Let us now give a couple of standard deﬁnitions and then rigorously deﬁne the static stellar model. We assume a metric signature of (−, +, +, +). Let Greek indices run from 0 to 3 and let Latin indices run from 1 to 3.

Energy-Momentum Tensor for a perfect ﬂuid:

Tµν = (ρ + p)uµuν + pgµν (1.1)

where u is a unit time-like vector field representing the 4-velocity of the fluid, ρ is the density and p is the pressure of the fluid.

Einstein Equation: 1 R = 8π(T − T λg ) (1.2) µν µν 2 λ µν where ,Rµν is the Ricci Tensor and gµν is the metric tensor.

A Static Stellar Model is a static, asymptotically flat space-time that satisfies the Einstein equation coupled with a perfect-fluid matter model. Physically, “static” means that the metric is time independent and the star is non-rotating. This corresponds to the mathematical definition.[20] A space-time M is called static if and only if there exists a 3-dimensional manifold

Σ and a diﬀeomorphism Ψ : M → Σ × R such that

3 −1 (i.) Cx = Ψ ({x} × R) are time-like curves for all x ∈ Σ −1 (ii.) Σt = Ψ (Σ × {t}) are globally space-like hyper-surfaces for all t ∈ R

(iii.) Cx for all x ∈ Σ are tangent to a Killing vector ﬁeld k on M that is orthogonal to all Σt

The main objective of this thesis is expository. This is a compilation of the work of brilliant men over the last 50 years. We hope to give a complete picture of existence and uniqueness of the static stellar model that is current to date. Simply put, the “Fluid Ball Conjecture” pertains to the actual shape of a star. It seems intuitively obvious that a highly idealized star modeled as a perfect fluid in equilibrium which does not rotate or change over time would be spherical in shape. In the context of General Relativity proving this expectation is not straight forward. This paper deals with this problem as a whole. We give a complete proof of the existence of spherically symmetric stellar models with an equation of state of acceptable regularity. It is a rather straightforward proof that utilizes a theorem from the theory of ordinary differential equations (O.D.E) which handles the singularity that arises in the center of the star. We also give a complete proof of the most recent result on the uniqueness of a static stellar model given by Masood ul-Alam. This proof constructs a conformal factor as a limit of constructed conformal factors and shows the spinor norm weighted scalar curvature integral goes to zero in the limit, implying in this case conformal flatness. Finally, we analyze constraints on certain physical quantities that occur in the framework of General Relativity. These constraints lend themselves as support to the “Fluid Ball Conjecture”.

4 Chapter 2

Existence and Uniqueness of Static, Spherically Symmetric Solutions

Existence and uniqueness proofs for spherically symmetric static stellar models with perfect ﬂuid source are scattered throughout literature. Lindblom and ul-Alam [29] proved existence in their joint work for a given equation of state, mass, and surface

potential Vs. Pfister [5] proved a general existence theorem for a certain class of equations of state using a Banach fixed point method. Mak and Harko [6] proved an existence theorem using a Riccati type first order O.D.E. with a solution expressed in the form of an infinite power series. The goal of this section is to discuss an existence theorem given by Rendall and Schmidt [7]. Although not the most general existence theorem for the spherically symmetric static stellar model with perfect fluid source, the proof is rather straightforward. We hold fast to the intuitive geometry of standard coordinates and the result follows from a modified existence theorem for singular ordinary differential equations. We start with a given central pressure and prove global existence and uniqueness for the Einstein equations representing the spherically symmetric static stellar model. Since we start from the center of the star,

5 it is possible that the star’s radius is infinite. In this case the vacuum region will be empty. If the star has a finite radius the boundary will occur at r = R where p(R) = 0 and p denotes the pressure. This particular existence and uniqueness theorem allows for stars of infinite radius, i.e. where the pressure does not have compact support. Certain results on the finiteness of the star can be derived from the given equation of state. We discuss this in a different section. We now state the main theorem.

Theorem 2.1 [Rendall and Schmidt (1991)] Let an equation of state ρ(p) be given such that ρ is deﬁned for p ≥ 0, non-negative,

∞ dρ and continuous for p ≥ 0, C for p > 0 and suppose that dp > 0 for p > 0.

Then there exists for any value of the central density ρ0 a unique inextendible static, spherically symmetric solution of Einstein’s field equations with a perfect fluid source and equation of state ρ(p). The matter either has finite extent, in which case a unique Schwarzschild solution is joined on as an exterior field, or the matter occupies the whole of space, with ρ tending to zero as r tends to infinity.

We note that the constraint of an equation of state ρ(p) being C∞ is one of convenience. This proof works for equations of state with lesser regularity. We now want to set the problem up.

2.1 Derivation of the System of Equations

The metric in Schwarzschild coordinates for a static, spherically symmetric space-time is given by

ds2 = −c2ea(r)dt2 + eb(r)dr2 + r2(dθ2 + sin2 θdφ2) (2.1)

6 where c represents the speed of light in vacuum and b, a are functions that only depend on r, the “area radius”. Notice that we use the metric signature (−, +, +, +). The Einstein ﬁeld equations are given by

1 8πG G = R − Rg = T (2.2) µν µν 2 µν c2 µν

where µ, ν = 0, 1, 2, 3, Gµν denotes the Einstein tensor, Rµν is the Ricci curvature, µ R = Rµ is the scalar curvature, and G is the gravitational constant. We take the cosmological constant Λ to be zero. The perfect ﬂuid stress energy tensor is given by

p T = ρu u + (u u + g ) (2.3) µν µ ν c2 µ ν µν

where u = (u0, u1, u2, u3) represents the components for the 4-velocity of our static √ a ﬂuid. Aligning the 4-velocity with the static Killing ﬁeld, we have u0 = − e and

since the ﬂuid is in equilibrium the spatial components are all zero, i.e. ui = 0 for i = 1, 2, 3. This gives us

νµ g uµuν = −1 (2.4)

Other variables include ρ, which is the proper energy density, and p, the proper pressure. Combining equations (2.1)-(2.3) we are able to derive the system of ﬁeld equations with respect to the above coordinates.[8]

8πG rb0 − 1 ρr2 = + 1 (2.5) c2 eb 8πG ra0 + 1 pr2 = − 1 (2.6) c4 eb 8πG 1 1 a0 − b0 1 p = (a00 + (a0)2 + − a0b0) (2.7) c4 2eb 2 r 2 where 0 denotes diﬀerentiation with respect to r. Also, we note that the ﬁeld equations for G22 and G33 are the same. This provides us with three independent equations and four functions. We follow a strategy outlined in [8]. We want to eliminate p from

7 equations (2.6) and (2.7). Setting these two equations equal and solving gives us the expression

eb a0b0 (a0)2 1 a0 + b0 1 = − − a00 + + (2.8) r2 4 4 2 2r r2 Next, we add equations (2.5) and (2.6) together, which gives us

8πG p a0 + b0 (ρ + ) = (2.9) c2 c2 reb We now want to divide both sides of (2.6) by r2 for r 6= 0 and diﬀerentiate both sides with respect to r. We have the expression

8πG 2 a0 a0b0 2 b0 a00 p0 = − e−b[ + + + − ] (2.10) c4 r3 r2 r r3 r2 r We can eliminate a00 from equation (2.10) by using equation (8). Rearranging terms and simplifying gives us the expression

8πG a0 (a0 + b0) p0 = − (2.11) c4 2 reb We combine equation (2.11) with (2.9). This gives us

p 2p0 = −c2a0(ρ + ) (2.12) c2 This equation represents the conservation of energy-momentum for a static perfect ﬂuid. Equations (2.5),(2.6),and (2.12) now contain all of the constraint information for our functions. We have only three equations for four functions for our system. We close this system by specifying an equation of state, ρ(p).

Now, equation (2.5) can be rewritten as

8πG d r2ρ(r) = (r − re−b + const.) (2.13) c2 dr

8 Integrating both sides with respect to r gives us

8πG 8πG 1 r2ρ(r)dr = r − re−b =⇒ e−b = 1 − r2ρ(r)dr + const. (2.14) c2 ˆ c2 r ˆ

For our solutions we are seeking regular centers of spherical symmetry. In order to avoid a “conical singularity” in the metric at r = 0, we set the constant equal to zero.[16] This gives us the following expression

r −b(r) 8πG 1 2 e = 1 − 2 s ρ(s)ds (2.15) c r ˆ0 We recall the expression for the Newtonian mass given up to radius r by the expression

r m(r) = 4π s2ρ(s)ds (2.16) ˆ0 Its derivative with respect to r is given by

m0(r) = 4πr2ρ(r) (2.17)

We combine equations (2.15) and (2.16) to get an expression for e−b

2G m e−b = 1 − (2.18) c2 r This is the spatial metric potential. Now, if we diﬀerentiate equation (2.18) with respect to r we get

2G rm0 − m e−bb0 = (2.19) c2 r2 and inserting equation (2.17) into (2.19) gives us

2G 4πr3ρ − m e−bb0 = (2.20) c2 r2

9 If we solve equation (2.9) for a0e−b we get

a0 8πG p = (ρ + )r − e−bb0 (2.21) eb c2 c2 We insert (2.20) into (2.21) and we get the expression

8πG p 2G (4πr3ρ − m) a0e−b = (ρ + )r − (2.22) c2 c2 c2 r2 8πG 8πG 8πG 2Gm = ρr + pr − ρr + (2.23) c2 c4 c2 c2r2 2 = (4πGpr3 + Gm) (2.24) c2r2

Solving for a0 we get

2 a0 = (4πGpr3 + Gm) (2.25) c2r2e−b If we insert the expression for a0 into the equation for energy-momentum conservation, which is equation (2.12) we get

2 p 2p0 = −c2( (4πGpr3 + Gm))(ρ + ) (2.26) c2r2e−b c2 2 p = −( (4πGpr3 + Gm))(ρ + ) (2.27) r2e−b c2

Finally, if we insert the expression in equation (2.18) into (2.27) we have the following expression.

1 p(r) 1 p0(r) = − (4πGp(r)r3 + Gm(r))(ρ(r) + )( ) (2.28) r2 c2 2G m(r) 1 − c2 r Equation (2.28) is known as the Tolman-Oppenheimer-Volkoﬀ (T.O.V.) equation of hydrostatic equilibrium. We collect some of the equations that are standard in

10 deriving interior solutions for static, spherically symmetric perfect ﬂuid stellar models.

1 p(r) 1 p0(r) = − (4πGp(r)r3 + Gm(r))(ρ(r) + )( ) (2.29) r2 c2 2G m(r) 1 − c2 r m0(r) = 4πr2ρ(r) (2.30) 2G m(r) e−b(r) = 1 − (2.31) c2 r p(r) 2p0(r) = −c2a0(r)(ρ(r) + ) (2.32) c2

We introduce an equation that represents a normalized mean density. It is given by

m(r) w(r) = (2.33) r3 Inserting this into equation (2.18) gives us

2G e−b = 1 − r2w(r) (2.34) c2 We can express the T.O.V. equation in terms of w using equations (2.33) and (2.34) which yields

1 p 1 p0 = − (4πGpr3 + Gr3w)(ρ + )( ) (2.35) r2 c2 2G 2 1 − c2 r w 1 p = −Gr( )(4πp + w)(ρ + ) (2.36) 2G 2 c2 1 − c2 r w

If we diﬀerentiate (2.33) with respect to r and use equation (2.30) we get

11 r3m0(r) − 3m(r)r2 3m(r) w0(r) = =⇒ r3w0(r) + = m0(r) (2.37) r6 r = 4πr2ρ(r) (2.38)

(2.39)

Simplifying this expression we get

1 r3w0(r) + 3r2w(r) = 4πr2ρ(r) =⇒ w0(r) = (4πρ(r) − 3w(r) (2.40) r

If we are given an equation of state, ρ(p) then we can use it to integrate equation (2.2). This gives us

p(r) 2 a(r) = − dp (2.41) ˆ p p0 ρ(p) + c2 where p0 is the pressure at the center of the star.

Finally, equations (2.35) and (2.40) form a system of equations for the functions p(r) and w(r). If we solve this system, then we can determine b(r) from equation (2.34) and a(r) from equation (2.41). We collect the system here

1 p p0 = −Gr( )(4πp + w)(ρ + ) (2.42) 2G 2 c2 1 − c2 r w 1 w0(r) = (4πρ(r) − 3w(r) (2.43) r 1 eb(r) = (2.44) 2G 2 1 − c2 r w(r) p(r) 2 a(r) = − dp (2.45) ˆ p p0 ρ(p) + c2

12 The derivation of the T.O.V equation was motivated and executed by eliminating a0 from equations (2.6) and (2.12). Instead of eliminating a0 from (2.6) and (2.12) we could eliminate p. For this purpose we consider the following auxiliary functions [9].

2G y2(r) = e−b(r) = 1 − r2w(r) (2.46) c2 a(r) z(r) = e 2 (2.47)

x(r) = r2 (2.48)

The goal now is to express equations (2.6) and (2.12) in terms of the new variables just given and then combine the equations, eliminating p. For (2.6) we need an expression

da for dr . From (2.47) we get

2 ln(z(r)) = a(r) (2.49)

Diﬀerentiating both sides of (2.49) with respect to x gives us

1 dz da dr da 1 2 = = √ (2.50) z dx dr dx dr 2 x

da Solving (2.50) for dr gives us

da √ 1 dz = 4 x (2.51) dr z dx We also have

da 2 dz = (2.52) dx z dx We want to express ρ in terms of w and the auxiliary functions (2.46)-(2.48). Recall equation (2.43). For this we have the following derivation

13 4π ρ = ρ (2.53) 4π 1 3m 3m = [ − + 4πρ] (2.54) 4π r3 r3 1 3m 2r2 3m = [ + (4πρ − )] (2.55) 4π r3 2r2 r3 1 1 1 = [3w + 2r2 (4πρ − 3w) ] (2.56) 4π r 2r 1 dw 1 = [3w + 2r2 ] (2.57) 4π dr 2r 1 dw dr = [3w + 2x ] (2.58) 4π dr dx 1 dw = [3w + 2x ] (2.59) 4π dx

We want to express (2.6) using functions (2.46)-(2.48) and (2.51). We express (2.12) using functions (2.46)-(2.48), (2.52) and (2.59). This gives us the following two equations.

8πG 1 dz 2G p = 4y2 − w (2.60) c4 z dx c2 dp 1 dz p 1 dz 1 dw p = − (ρ + ) = − ( [3w + 2x ] + ) (2.61) dx z dx c2 z dx 4π dx c2

We want to now eliminate dependence on p from (2.60) and (2.61) and set the equation to equal zero. This equation can be expressed in the following way by means of calculation and simpliﬁcation [7], [9].

2G d2z G dz d dw G dw 0 = (1 − xw) − [ (w + x )] − z (2.62) c2 dx2 c2 dx dx dx 2c2 dx d2z dy dz G dw = y2 + y − z (2.63) dx2 dx dx 2c2 dx d dz G z dw = (y ) − (2.64) dx dx 2c2 y dx

14 If ρ is given then equation (2.62) is linear in the variable z and if z is given, then

dρ dw (2.62) is linear in w.[9] We next want expressions for dx and dx . Using equation (2.35) and r2 = x we get

dρ dp dp = ( )−1 (2.65) dx dρ dx dp dp dr = ( )−1 (2.66) dρ dr dx dp 1 p 1 = ( )−1[−Gr( )(4πp + w)(ρ + )] √ (2.67) dρ 2G 2 c2 1 − c2 r w 2 x G dp 1 p = − ( )−1( )(4πp + w)(ρ + ) (2.68) 2 dρ 2G c2 1 − c2 xw

and using equation (40) and r2 = x we get

dw dw dr = (2.69) dx dr dx 1 1 = (4πρ − 3w) √ (2.70) r 2 x 1 = (4πρ − 3w) (2.71) 2x

Equations (2.68) and (2.71) provide a system of equations for unknown functions w(x) and ρ(x). Solving for w allows us to recover b from (2.46). Given an equation of state, and having solved the system for ρ we then solve (2.45) for a. We collect this system here.

15 dρ G dp 1 p = − ( )−1( )(4πp + w)(ρ + ) (2.72) dx 2 dρ 2G c2 1 − c2 xw dw 1 = (4πρ − 3w) (2.73) dx 2x p(r) 2 a(r) = − dp (2.74) ˆ p p0 ρ(p) + c2 b 1 e = 2G (2.75) 1 − c2 xw

Finally, equations (2.72)-(2.75) represents the system our proof will deal with. First note that while equation (2.72) is regular at x = 0, (2.73) is not. In an eﬀort to have uniform properties for both equations we want to make (2.72) singular at x = 0.

Deﬁne a function ρ1 so that

ρ = ρ0 + xρ1 (2.76)

where ρ0 denotes the density at the center of symmetry. Similar to equation (2.77) we have the corresponding relationship

p(ρ) = p0 + xp1(ρ1) (2.77)

where p0 denotes the central pressure. Substituting (2.76) and (2.77) into equation (2.72),(2.73) and algebraically manipulating (2.73) gives us

dρ G dp 1 x 1 + ρ = − ( (ρ ) + xh(ρ ))−1( )(4πp + 4πxp (ρ ) + w) (2.78) dx 1 2 dρ 0 1 2G 0 1 1 1 − c2 xw p xp (ρ ) · (ρ + xρ + 0 + 1 1 ) 0 1 c2 c2 dw 3 x + w = 2πxρ + 2πρ (2.79) dx 2 1 0

dp dp where dρ = dρ (ρ0) + xh(ρ1) for some smooth function h.

16 2.2 The Singular Point

Rendall and Schmidt prove an existence theorem for singular ordinary diﬀerential equations. We state it now.

Theorem 2.2 [Rendall and Schmidt (1991)] Let V be a ﬁnite-dimensional real vector space, N : V → V a linear mapping, F : V × I → V a smooth (i.e. C∞) mapping and g : I → V a smooth mapping, where I is an open interval in R containing zero. Consider the equation

df s + Nf = sF (s, f(s)) + g(s) (2.80) ds

for a function f deﬁned on a neighborhood of 0 in I and taking values in V . Suppose that each eigenvalue of N has positive real part. Then there exists an open interval J with 0 ∈ J ⊂ I and a unique bounded C1 function f on J \{0} satisfying (2.80). Moreover f extends to a C∞ solution of (2.80) on J. If N,G and g depend smoothly on a parameter γ and the eigenvalues of N are distinct then the solutions also depends smoothly on γ.

We want to apply Theorem 2.2 to the set of equations (2.78) and (2.79). We ﬁrst

want to put it into the proper form. We ﬁrst deﬁne f : R → R2 by

  ρ1(x) f(x) =   (2.81) w(x)

We now want to look at equation (2.78). Note that we have the following equivalence.

2G 2G 2G (1 − xw)−1 = 1 + xw(1 − xw)−1 (2.82) c2 c2 c2 Inserting (2.82) into (2.78) gives us

17 dρ G dp 2G 2G x 1 + ρ = − ( (ρ ) + xh(ρ ))−1[1 + xw(1 − xw)−1](4πp + 4πxp (ρ ) + w) dx 1 2 dρ 0 1 c2 c2 0 1 1 (2.83) p xp (ρ ) · (ρ + xρ + 0 + 1 1 ) 0 1 c2 c2

We expand the right-hand side of (2.83) and collect terms. We then deﬁne functions

2 F1 : I × R → R and g1 : I → R where I ⊂ R is an open interval containing 0. This way we can write (2.83) as

dρ G dp p x 1 + ρ = − (ρ )( 0 + ρ )w + xF (x, f(x)) + g (x) (2.84) dx 1 2 dρ 0 c2 0 1 1 Further still, we write (2.84) as

dρ G dp p x 1 + ρ + (ρ )( 0 + ρ )w = xF (x, f(x))) + g (x) (2.85) dx 1 2 dρ 0 c2 0 1 1

2 Now, for equation (2.79) we deﬁne functions F2 : I × R → R and g2 : I → R by

F2(x, f(x)) = 2πρ1(x) (2.86)

g2(x) = 2πρ0 (2.87)

This allows us to express (2.79) as

dw 3 x + w = 2πxρ + 2πρ = xF (x, f(x)) + g (x) (2.88) dx 2 1 0 2 2

We can now deﬁne our function N : R2 → R2 in matrix form. It is given by

p0  G( +ρ0)  1 c2 2 dp (ρ ) N =  dρ 0  (2.89) 3 0 2

18 We can now write out our system of equations (2.78),(2.79) in the form given by Theorem 2.2. We use the expression in (2.85),(2.88), and (2.89). This gives us

df x + Nf(x) = xF (x, f(x)) + g(x) (2.90) dx

p0    G( +ρ0)        1 c2 d ρ1(x) 2 dp (ρ ) ρ1(x) F1(x, f(x)) g1(x) x   +  dρ 0    = x   +   dx 3 w(x) w(x) F2(x, f(x)) g2(x) 0 2 (2.91) We now just note that F is a combination of smooth functions, and hence smooth. g is a combination of smooth functions, and hence smooth. The matrix N is upper

3 triangular with eigenvalues 1 and 2 hence all its eigenvalues are real and positive. Therefore, by theorem 2 there exists an open interval J with 0 ∈ J ⊂ I and a unique bounded C1 function f on J \{0} satisfying (2.91). Moreover, f extends to a C∞ solution of (2.91) on the entire interval J. This means that we have a unique smooth solution of equations (2.72) and (2.73) in a neighborhood of 0 for the given central

density ρ0 and equation of state ρ(p).

2.3 Extending the Solution Uniquely

Our ﬁnal step in the proof is showing that our unique, smooth solution f can be

extended in a unique way. If our star is ﬁnite, there will be some xb such that

p(xb) = 0 where xb corresponds to R which is the radius of the star. In this case, the Schwarzschild vacuum solution will connect to the interior solution at the boundary region.[10] If the star is infinite, then p(x) > 0 for all x,ρ → 0 as x → ∞, and the vacuum region is empty. Standard existence and uniqueness theorems for ordinary differential equations imply that f can be extended in a unique way as long as the right-hand sides of (2.72) and (2.73) are well-defined.[7]. We first look at (2.72).

19 dρ We want to make sure that as x → ∞ dx > 0 for all x in the interior solution, i.e. the region where p > 0. If the star is inﬁnite then the “interior” solution will dρ occupy all of space. Our hypothesis for Theorem 2.1 includes dp > 0 and ρ(p) ≥ 0 for p ≥ 0. We also know that G, c > 0 and w > 0 for p > 0. These imply that the ﬁrst, second, and fourth factors in equation (2.72) are positive in the “interior” region.

2 2G We ﬁrst want to show that y = 1 − c2 xw cannot vanish while p > 0. If this were to occur, then p > 0 implies we are still in the “interior” solution. If y2 vanished dρ in the interior, then dx = 0 and we would lose uniqueness of our extension. Let 2 dρ 0 ≤ x < xb be an interval where y > 0 and p > 0. We know that since dp > 0 that the density ρ(p) does not increase outward, i.e. when p decreases. This implies that

dw dx ≤ 0. Equation (2.64) then gives us

d dz G z dw (y ) = ≤ 0 (2.92) dx dx 2c2 y dx Since z, y > 0 and G, c are positive constants. Equation (2.60) can be rewritten as

dz G z 4πp y = [ + w] (2.93) dx c2 y c2

dz Equation (2.92) gives us that the derivative of y dx is non-positive, which means that the expression in (2.93) is non-increasing. Therefore we have

dz dz y ≤ y | (2.94) dx dx x=0 This gives us the following inequality, recalling (2.46)-(2.48).

20 G z 4πp G 4πp [ + w] ≤ [ c + w ] (2.95) c2 y c2 c2 c2 0 z 4πp 4πp [ + w] ≤ c + w (2.96) y c2 c2 0 [ 4πp + w] z c2 ≤ y (2.97) 4πpc c2 + w0 [ 4πp + w] c2 ≤ y (2.98) 4πpc c2 + w0

Therefore, (2.98) shows that it is impossible for y to vanish before p. This of course

2 2G dρ implies that y = 1 − c2 xw cannot vanish in the region where p > 0. Hence dx > 0 for p > 0 and ρ extends uniquely to the boundary of the star. For equation (2.73) we recall the expression in (2.79) and note that ρ = ρ0 + xρ1 is unique and smooth up to the boundary and so w must also be smooth and unique up to the boundary.

Consider the case where the star is ﬁnite. The radius is given by R which

corresponds to some xb. The exterior solution is the Schwarzschild solution and is uniquely given by the following metric potentials.

2G m(R) ea(r) = e−b(r) = 1 − (2.99) c2 r The spatial metric potential for the interior solution is given by equation (2.75) by the following.

2G 2G m(r) e−b(r) = 1 − r2w(r) = 1 − (2.100) c2 c2 r So we see by equation (2.99) and (2.100) that the spatial potential for the interior solution matches up to the spatial potential of the exterior vacuum solution at the boundary, r = R. For the space-time potential in the interior, we have by equation

21 (2.74) that p(r) 2 ea(r) = exp[− dp] (2.101) ˆ p p0 ρ(p) + c2 at the boundary, r = R, the space-time potential can only be joined together continuously, i.e. in C0 fashion. This is because it may be the case that ρ(R) > 0. If ρ does not vanish at the boundary then we can get at most a C1 space-time potential. This follows from equation (2.2) and the fact that if ρ does not equal zero at the boundary then the Ricci tensor must be discontinuous at the boundary. We also make note that the metric cannot be extended because as r increases so does the area of the r = const. group orbits.

In the case of an inﬁnite star, we have p(r) > 0 for all r > 0. We know that limr→∞ p(r) exists since p(r) is bounded below by 0 and monotonically decreasing. dp 2 This implies that dr must vanish at inﬁnity. Recall equation (2.46). Note that y ≤ 1. (2.42) give us

dp 1 p = −Gr (4πp + w)(ρ + ) (2.102) dr y2 c2 We know w vanishes at inﬁnity by deﬁnition since m(r) ≤ m(R) < ∞. Also, ρ ≥ 0

dp 1 for p ≥ 0 The fact that limr→∞ dr = 0, −Gr → −∞ as r → ∞ and y2 ≥ 1 gives us

dp 1 p 0 = lim = − lim Gr (4πp + w)(ρ + ) (2.103) r→∞ dr r→∞ y2 c2 p ≤ − lim Gr(4πp + w)(ρ + ) (2.104) r→∞ c2

Clearly we see that limr→∞ p(r) = 0. This implies that limr→∞ ρ(r) = 0. This completes the proof of Theorem 2.1.

22 Chapter 3

Uniqueness of the Static Stellar Model

The metric tensor in a static model in General Relativity has the form:

2 2 2 a b ds = −V dt + gabdx dx (3.1)

where gab, a, b = 1, 2, 3, denotes the spatial metric for the t = const. hyper-surfaces

and V and gab are time-independent. Coupling Einstein’s equation with a perfect ﬂuid matter model gives us the following pair of equations [31]

a D DaV = 4πV (ρ + 3p) (3.2)

−1 Rab = V DaDbV + 4π(ρ − p)gab (3.3)

where the density and pressure are denoted by ρ and p, respectively. Da and Rab are with respect to the spatial metric, and the density is assumed to be a given function of pressure, ρ(p), which is referred to as the equation of state. Combining the two equations above and the Bianchi identity yields:

−1 Dap = −V (ρ + p)DaV (3.4)

23 If this static stellar model was spherically symmetric, then this equation would be equivalent to the so-called Tolman-Oppenheimer-Volkoﬀ (T.O.V.) equation for hydrostatic ﬂuids for a metric tensor of the above form. These equations have become standard over the evolution of the attempt to solve this problem.

The idea of using the Positive Mass Theorem to prove rotational symmetry of static stellar models which we analyze here was introduced by Masood ul-Alam in the late 1980’s. Slight variations have been used in order to obtain uniqueness while lessening the restrictions on the acceptable equations of state. The scheme is to first parametrize the system in terms of the potential V , where the surface potential Vs represents the boundary of the star. The task is then to derive a conformal factor which transforms the spatial metric of the t = const. hyper-surface of the static stellar model into a metric with zero mass and non-negative scalar curvature. The rigidity part of the Positive Mass Theorem then implies that the conformal metric must be flat. An old result due to Avez [23] says that a spatially conformally flat static perfect- fluid solution is necessarily spherically symmetric. The technical difficulties with this method revolve around showing the that the scalar curvature of the conformal metric is non-negative. The scalar curvature associated with conformal metric parameterized by V is given by the equation

8 d2ψ Rb = (Wf − W ) (3.5) ψ5 dV 2

a where W = D DaV is the square of the field intensity for the static stellar model, ψ is the conformal factor which transforms the spatial metric, and Wf is the norm squared of the field intensity for a “reference model”. More precisely, Wf is related to ψ via the second-order linear differential equation:

d2ψ dψ Wf (V ) = 2π[ρψ − 2V (ρ + 3p) (V )] (3.6) dV 2 dV

24 In general, the conformally transformed scalar curvature is related to the original scalar curvature by the formula

−4 −1 a Rb = ψ (R − 8ψ D Daψ) (3.7)

where the scalar curvature R and the covariant derivative Da are with respect to the spatial metric.[32]. The variations on this overall prescription have revolved around choosing ψ and Wf in a clever way. Outside the boundary of our ﬁnite static star, the metric is necessarily Schwarzschild, which follows from the fact that outside of our star we have a vacuum solution of Einstein’s equation. In the case rotational symmetry holds (as we hope to prove), Birkhoﬀ’s Theorem then implies that the metric is Schwarzschild. Therefore, outside of the star, we choose the conformal factor 1 ψ = (1 + V ) (3.8) 2 for V ≥ Vs. (That his corresponds to the Schwarzschild metric is easily shown, see [31].) Using this conformal factor the scalar curvature with respect to the conformal metric outside of the star will be zero. The mass of the conformal metric will also be zero. What is left is to determine Wf and ψ for the interior solution. Choosing Wf and ψ for the interior solution is what directs us to add restrictions to the equation of state. The challenge of guaranteeing the sign of Rb in the interior of the star be non-negative is the heart of the variations of this method.

Lindblom in his 1988 paper [22] dealt with constant density models (see also [31]). This allowed the equation of hydrostatic equilibrium to be integrated explicitly. He then assumed without proof the existence of a “reference spherical model” which possesses the same equation of state, in this case the same constant density, and the same surface potential Vs. This “reference spherical model” is then used to deﬁne ψ 8 d2ψ and Wf inside the star. This leads to ψ5 dV 2 being non-negative inside the star and

25 vanishing on the outside. The sign of (Wf − W ) in this case can then be determined using Robinson type identities and the maximum principle for elliptic operators.

In 1993 [29], Lindblom and ul-Alam demonstrated the existence of a “reference

spherical model” for an appropriately chosen mass parameter ν = M(Vs) which possesses the same equation of state and surface potential value as the static stellar

model. This means ﬁnding ‘area radius’ and ‘mass’ functions rν(V ), mν(V ), solving a system of ordinary diﬀerential equations described below, and from them computing

the ‘field strength’ function Wν(V ). The conformal factor is then found inside the star to by solving the first-order linear differential equation

dψ ψ r 2m = √ν [1 − 1 − ν ] (3.9) dV 2rν Wν rν

1 with the boundary condition ψν(Vs) = 2 (1 + Vs). The parameters subscripted by ν are the parameters of the “reference spherical model”. The significance of this equation is that in the case of a spherically symmetric model with a mass parameter of ν, the conformal factor which transforms the spatial metric to a flat spatial metric satisfies this equation [31]. Unique solutions rν(V ) and mν(V ) exist (inside the star) if we assume that the equation of state is at least C1. Specifically, the equation

1 1 of state being C implies that ρ(V ) and p(V ) are also C for V < Vs, and then standard existence-uniqueness for O.D.E. systems give solutions on a maximal domain

(Vν,Vs].[29] In this case we set Wf = Wν. One of the pivotal points of this variable mass modiﬁcation being successful is monotonicity with respect to the mass parameter

µ of the function Wµ(V ) for a ﬁxed V ∈ (Vµ,Vs). This monotonicity result follows from showing that the sign of the expression

dW 8π 4W ρ + p dρ Σ = µ − V (ρ + 3p) + µ (3.10) µ dV 3 5V ρ + 3p dp

26 is non-negative. The sign of Σµ is guaranteed to be non-negative if the equation of state satisﬁes one of two conditions, Condition A or Condition B. Assuming that the equation of state is at least C1 and satisﬁes either Condition A or Condition B ensures that Σν is non-negative. This implies that Wµ is monotonic with respect

to µ. Monotonicity then implies that for Wf = Wν, we have Wf − W ≥ 0. For 8 d2ψ the quantity ψ5 dV 2 to be guaranteed non-negative the equation of state must satisfy another condition 5ρ2 − 6p(ρ + 3p)κ ≥ 0 (3.11)

ρ+p dρ where κ = . If it happens to be the case that lim + r > 0 for the “reference ρ+3p dp V →Vν ν spherical model”, then a slight perturbation is made in the mass parameter ν. ψ and Wf are then chosen based on the perturbed mass ν + δ. The two factors of the conformal scalar curvature seem to not be directly related. However, the conditions on the equation of state are. In fact, the inequality just mentioned is exactly Condition B. Moreover, it is shown that Condition A implies Condition B. So an equation of state which satisﬁes either Condition A or Condition B will imply that both factors of the conformal scalar curvature are non-negative. Then the line of argument described above implies the result. Uniqueness, in fact, implies that the mass parameter ν is actually equal to the ADM mass M of the given static stellar model.

The most recent uniqueness proof by ul-Alam [4] takes a more localized approach. The basic outline of his strategy is the same. The biggest modiﬁcation made by ul- Alam is by utilizing the spinor norm weighted scalar curvature integral that appears in Witten’s proof of the Positive mass theorem. This proof moves away from trying to construct a conformal factor that forces the point-wise non-negativity of the conformal scalar curvature and instead constructs a conformal factor such that the negative contribution of the conformal scalar curvature to the spinor norm weighted integrated scalar curvature can be made as small as we like. This relaxes the need for the scalar curvature of the conformal metric to be non-negative everywhere. We allow areas of

27 the conformal geometry to have negative scalar curvature but the contribution to the integral must be small. Instead of deﬁning a global conformal factor and a global Wf he constructs a sequence of conformal factors and a corresponding sequence of Wf

starting at V = Vs and working inward. This approach entails many more technical details but strengthens the overall result in that the only restriction on the equation of state is that it is piecewise C1. The remainder of this section will be devoted to outlining the proof of ul-Alam’s result which constitutes the most recent uniqueness proof for the static stellar model. We ﬁrst give the statement of the theorem and list the hypothesis.

28 3.1 Statement of the theorem.

Theorem 3.1 [M. ul-Alam (2007)] A static stellar model satisfying the following assumptions is necessarily spherically symmetric. i.) The space-time 4-manifold is M 4 = N 3 × R with line element

2 2 2 i j ds = −V dt + gijdx dx gij is a complete Riemannian metric on N and V : N → [Vmin, 1). ii.) The spatial metric and gravitational potential satisfy the asymptotic conditions:

2m g = (1 + )δ + O(r−2) ij r ij m V = 1 − + O(r−2) r iii.) The density ρ = ρ(p) is a piecewise C1 positive, non-decreasing function of p for p > 0. ρ = 0 in the exterior region. vi.) The sets on the spatial hyper-surface N along which ρ has discontinuity are smooth 2-surfaces. There are at most a ﬁnite number of these surfaces. v.) The pressure p = p(r) is globally Lipschitz and p > 0 in the ﬂuid region and p = 0 in the exterior vacuum. p is a non-negative, bounded, measurable function.

vi.) The boundary V = Vs < 1 of the interior ﬂuid region and the vacuum region are both level sets of V . The level set of Vs is a smooth 2-surface.

29 1,1 3 vii.) The gravitational potential V and the metric gij are C globally and locally C in the complement of the smooth 2-surfaces where ρ has discontinuity and the level

set of Vs.

The Einstein equations for a static Lorentzian metric:

2 2 2 i j ds = −V dt + gijdx dx (3.12)

with the perfect ﬂuid energy-momentum tensor reduce [31] to the following system of equations:

i D DiV = 4πV (ρ + 3p) (3.13)

−1 Rij = V DiDjV + 4π(ρ − p)gij (3.14)

where Di and Rij are with respect to the Riemannian 3-metric, gij. The diﬀerential

Bianchi identity for gij implies the equation for hydrostatic equilibrium

−1 Dip = −V (ρ + p)DiV (3.15)

We carry out the integration of the equation for hydrostatic equilibrium on an interval

1 [V,Vs] where ρ is a C function of p which gives us

V p ds ln( s ) = = h(p) (3.16) V ˆ0 ρ(s) + s

The right hand side of this equation is invertible so we can consider p and ρ as functions of the potential

V p(V ) = h−1(ln( s )) (3.17) V ρ(V ) = ρ(p(V )) (3.18)

30 1 1 Since ρ(p) is C in this interval, it follows that p(V ) and ρ(V ) are also C for V < Vs which follows from the fact that V is assumed to be C3 away from the boundary and on the complement of the discontinuities of ρ.

3.2 The “reference system” of O.D.E.

We state now a local existence lemma of the spherically symmetric equations from ul-Alam and Lindblom’s joint work.[29]

Consider the following system of equations.

dr r(r − 2m) = dV V (m + 4πr3p) dm 4πr3(r − 2m)ρ = dV V (m + 4πr3p)

1 Here ρ(V ), p(V ) are given C functions in [Vm, b], and initial conditions are given at 3 V = b, satisfying r > 0, r > 2m > −8πr p. (Where b ∈ (Vm,Vs] is arbitrary). For a spherically symmetric solution (with metric coeﬃcients depending only on V ) the

i squared ﬁeld strength DiVD V admits the expression:

2m dr V 2(m + 4πr3p)2 Wf = (1 − )( )−2 = . r dV r3(r − 2m)

Let (Vc, b] be the maximal interval of existence for a solution with given initial conditions r(b), m(b). Then r > 0, r > 2m > −8πr3p and Wf > 0 on this interval, and furthermore we are assuming p(V ) < ∞ for V ∈ [Vm, 1) (where Vm is the minimum value of V in the interior).

31 Lemma 3.1 [ul-Alam (2007)]

Either the maximal interval of existence (Vc, b] contains [Vm, b], or Vc ∈ [Vm, b). In either case, r and m are monotonic functions of V and sup[Vc,b](2m/r) < 1 on the corresponding interval ([V , b] or (V , b]). In the second case lim + W = 0. Then m c V →Vc f either lim + r(V ) = 0 (“regular zero” of W ) or lim + r(V ) > 0 (“irregular zero”). r→Vc f r→Vc −3 −3 If the former, we have mr → (4π/3)ρ(Vc) as V ↓ Vc; if the latter, mr → −4πp(Vc). In particular, m < 0 on approach to an irregular zero of Wf.

3.3 Spinor approach to the Positive Mass Theo- rem.

In order to understand the strategy of our proof it is necessary to consider Bartnik’s version [17] of Witten’s Positive Mass Theorem[30]. We state it here.

Theorem 3.2 [ Bartnik (1985)] Suppose that (M n, g) is a complete spin manifold satisfying the asymptotic ﬂatness conditions:

2,q i.) (Φ∗g − δ) ∈ W−τ (ER0 ) for some asymptotic structure Φ, R0 > 1, q > n, (3.19) 1 and τ ≥ (n − 2) 2 ii.) R(g) ∈ L1(M n, g) (3.20)

with non-negative scalar curvature: R(g) ≥ 0. Let ξ0 be a spinor, constant near 2 inﬁnity and normalized by |ξ0| → 1 at inﬁnity, and let ξ be the unique solution of Dirac’s equation satisfying: 2,q ξ − ξ0 ∈ W−τ .

32 Then the mass of M n is non-negative and is given by

2 2 c(n)mass(g) = (4|∇ξ| + R|ξ| )dvolg (3.21) ˆM

Furthermore, if mass(g) = 0, then M is ﬂat. (Here c(n) = 2(n − 1)ωn−1.)

A desirable approach along the lines of ul-Alam’s method for a uniqueness proof

4 would be to construct a conformal factor ψe which makes mass(gb) = 0 (where gb = ψe g) and use the theorem above. We do in fact have a spin manifold since any orientable 3-manifold has a spin structure.[13] The problem is that we do not know the sign of Rb at this point. The sign of Rb is necessary in proving that vanishing of the mass implies ﬂatness.

It is important now to recall the argument used by Bartnik to deﬁne his mass integral and to conclude that the metric was ﬂat in the case that the mass of the metric was zero. The key to Witten’s method is the Lichnerowicz-type identity

1 (|∇ψ|2 + R|ψ|2 − |Dψ|2) ∗ 1 = d(< ψ, σ .∇ ψ > ∗e ) (3.22) 4 ij j i

where D is the Dirac operator and

1 σ = [e , e ] = e e + δ (3.23) ij 2 i j i j ij

(Here (ei) is a local orthonormal frame, and we use Clifford multiplication.) Its derivation can be found in the appendix of this paper. The goal was to find an asymptotically constant spinor field, ψ, satisfying Dψ = 0, and then identify the right hand side of Witten’s identity, (3.22), with the mass. The first step that Bartnik took was to show that the Dirac operator was in fact an isomorphism between certain weighted Sobolev spaces. This was Proposition 6.1 in [17]. In proving that these the

33 two weighted Sobolev spaces were isomorphic non-negative scalar curvature was used in conjunction with the strong maximum principle to show that the kernel of D and

its adjoint was trivial. Secondly, it was shown that for an arbitrary spinor field ψ0, which is constant at infinity, there exist a spinor field ψ which satisfies Dψ = 0 and that ψ −ψ0 are elements of a weighted Sobolev space where the weight is equal to the rate of the mass decay condition for the manifold. This was given in Corollary 6.2 in [17]. This shows existence of the spinor occurring in Bartnik’s integral expression for the ADM mass, after identifying the boundary term in Witten’s identity,(3.22), with the ADM mass. The positivity of mass is then an easy consequence assuming the non-negativity of the scalar curvature. Finally, in the case that the mass was zero we have c(n)mass(g) = (4|∇ψ|2 + R|ψ|2) ∗ 1 = 0 (3.24) ˆM

This implies that ∇ψ ≡ 0. From the spinor ψ we can deﬁne a vector vψ via the surjective map

n < vψ, X >= Im < ψ, X.ψ > for X ∈ R (3.25)

Since ∇ψ = 0 then ∇vψ = 0. Since ψ0 was an arbitrary constant spinor at infinity in our construction we can find a basis for TM consisting of covariantly constant vector fields. This means that M must be flat.

3.4 Strategy of proof of the Main Theorem.

The importance of outlining Bartnik’s proof is that we will use the same technique in order to prove that with a properly defined conformal factor we too will have a flat metric for our static stellar model. Namely, by choosing an appropriate conformal factor, we can make the spinor norm weighted scalar curvature as small as we like. Taking limits, we find three linearly independent parallel spinors, which allows us to

34 find a basis for TM consisting of covariantly constant vector fields. We use this to conclude that the conformal metric is flat.

4 We denote the scalar curvature of the conformal metric gb = ψe g by Rb, given by the following equation: 8 d2ψe Rb = (Wf − W ) 2 (3.26) ψe5 dV

(See (3.6) for Wf.) The sign of Rb is not yet known at this point so we cannot apply Bartnik’s argument for existence of such a spinor with respect to our conformal metric

gb and hence cannot yet deﬁne mass(gb). However, an easy calculation shows that the scalar curvature R with respect to our original metric g is given by

R = 16πρ (3.27)

Since ρ ≥ 0 we know that R ≥ 0. Therefore, we can apply Bartnik’s argument for

the existence of the needed Dirac spinor with regards to the metric g. Let ξ0 be an arbitrary spinor ﬁeld that is constant at inﬁnity. Corollary 6.2 from Bartnik [17] then

says that there exists a spinor ﬁeld ξ = ξ0 + ξ1 which satisﬁes Dξ = 0 and decays

to ξ0 at the needed rate. In other words, ξ is constant at inﬁnity and ξ1 falls oﬀ to −τ 1 the order of O(r ) for some τ > 2 . Therefore the mass of our metric g admits the expression:

2 2 c(n)mass(g) = (4|∇ξ| + R|ξ| )dvolg (3.28) ˆM

We want to deﬁne a conformal factor ψe(V ) for V ∈ [Vm, 1). Outside of our static star Birkhoﬀ’s theorem says that our space-time should be Schwarzschild. In this case, we know the form of the conformal transformation which sets the mass and scalar curvature to zero. The conformal transformation is

1 ψ(V ) = (1 + V ) (3.29) 2

35 for V ∈ [Vs, 1). We start deﬁning ψe by setting ψe = ψ for V ∈ [Vs, 1). Therefore,

Rb = 0 on [Vs, 1) and the mass of gb is zero. Under a conformal transformation the spinor Θ = ψe−2ξ satisﬁes the Dirac equation DΘ = 0 relative to the conformal metric gb.[3] The spinor Θ is given by

−2 −2 Θ = ψe ξ = ψe (ξ0 + ξ1) = ξ0 + Θ1 (3.30)

−2 −τ Θ1 = ψe ξ1 will fall off like O(r ) which follows from our assumptions on ξ1 and the conformal factor approaching a constant at infinity. Using the spinor Θ with its falloff we can identity the right hand side of Witten’s identity with the mass of gb just as Bartnik did in his proof. Since the mass of gb is zero, the integral formula with respect to gb becomes (RkΘk2 + 4k∇ Θk2)dvol = 0 (3.31) b gb gb ˆM We express the first term with respect to the original metric g. This gives us

0 = (RkΘk2 + 4k∇ Θk2)dvol = Rψ2kξk2dvol + 4k∇ Θk2dvol . (3.32) b gb gb b e g gb gb ˆM ˆM ˆM

We will divide [Vm,Vs] into two sets, A and B. On the set A we deﬁne ψe to satisfy the second order linear O.D.E.:

d2ψe 2π dψe 2 = [ρψe − 2V (ρ + 3p) ] (3.33) dV Wf dV assuming Wf > 0 is given. The signiﬁcance of the second order ODE is as follows. In the case that the stellar model is spherically symmetric Einstein’s equation coupled with the perfect ﬂuid matter model will yield the following equations

dr r(r − 2m) = (3.34) dV V (m + 4πr3p) dm 4πr3(r − 2m)ρ = (3.35) dV V (m + 4πr3p)

36 and then the squared ﬁeld intensity is:

2m dr V 2(m + 4πr3p)2 W (V ) = (1 − )( )−2 = (3.36) r dV r3(r − 2m)

Solutions for these equations are given for V ∈ [a, b] with initial values given at V = b. Also, for a stellar model which is spherically symmetric the conformal factor ψe which makes ψe4g ﬂat will satisfy the linear ODE

r dψe ψe 2m = [1 − ] (3.37) dV p r 2r Wf

Diﬀerentiating this whenever possible will yield the second order O.D.E., (3.33). On the set B we deﬁne ψe = u where u is a function for which we have control over the scalar curvature of u4g. If we break the integral up along the sets A and B, where

A ∪ B = [Vm,Vs], and recall that for V ∈ (Vs, 1] we have Rb = 0 we get

0 = Rψ2kξk2dvol + 4k∇ Θk2dvol (3.38) b e g gb gb ˆM ˆM 2 2 2 2 2 2 = Rbψe kξk dvolg + Rbψe kξk dvolg + Rbψe kξk dvolg −1 −1 −1 ˆV (A) ˆV (B) ˆV ((Vs,1)) (3.39)

+ 4k∇ Θk2dvol gb gb ˆM = Rψ2kξk2dvol + Rψ2kξk2dvol + 4k∇ Θk2dvol (3.40) b e g b e g gb gb ˆV −1(A) ˆV −1(B) ˆM 2 8 d ψe 2 2 2 2 4 = (Wf − W ) 2 ψe kξk dvolg + Ru gu kξk dvolg+ (3.41) ˆV −1(A) ψe5 dV ˆV −1(B) 4k∇ Θk2dvol gb gb ˆM 2 8 d ψe 2 2 2 2 = (W − W ) kξk dvol + R 4 u kξk dvol + 4k∇ Θk dvol f 2 g u g g gb gb ˆV −1(A) ψe3 dV ˆV −1(B) ˆM (3.42)

37 From this vantage point our ultimate goal can be seen. We want to be able to make the third integral as small as we like. What is left is to deﬁne the sets A and B, the function ψe and Wf on A and u on B.

We want to look at the integral in equation (3.42) and construct the functions

Wf, ψe, and u on intervals contained in [Vm,Vs] depending on what set the interval d2ψe falls in and the behavior of dV 2 at the right endpoints. First, we have a lemma gives

existence of solutions for the spherically symmetric ODE system near Vs and starting

properties for the solutions at V1 ∈ [Vm,Vs).

Lemma 3.2 [ul-Alam (2007)]

There exists a noncritical value V1 < Vs, and solutions (r, m, W,f ψe) of equations

(3.34)-(3.37) on [V1,Vs] with ψe > 0, and if for V ≥ Vs we deﬁne ψe as in (3.29), then ψe 1,1 d ln ψe d2ψe is C on [V1, 1). Furthermore on (V1,Vs] 0 < 2 dV < 1, Wf −Wave > 0, and dV 2 > 0. 8 d2ψe 2 In particular, −1 (Wf − W ) 2 kξk dvolg > 0. (Wave is deﬁned later.) V ([V1,Vs]) ψe3 dV ´

3.5 Auxiliary system of diﬀerential equations.

Lemma 3.2 gives us positivity of the integral on the set [V1, 1) and the starting d2ψe conditions of Wf and dV 2 . Our goal is to continue into the star constructing the conformal factor ψe. Some intervals we will use Wf to deﬁne ψe by means of equation

(3.33). On certain intervals in [Vm,V1], however, we will define ψe according to the solution to one of two differential equations. On these certain intervals, which we will define later, we want ψe to have certain properties which depend on the properties of the interval it is defined on. So on certain intervals we will set ψe = u where u is the solution to this DE we not describe.

38 Let γ denote the adiabatic index

ρ + p dp γ := (3.43) p dρ

dρ We know that for our model star dp ≥ 0 which means that γ > 0 on the interior of

the star, V < Vs. A simple computation yields the next lemma.

Lemma 3.3 [ul-Alam (2007)]

d2ψe At V < Vs where dV 2 = 0 and ψe is three times diﬀerentiable we have

3 2 d ψe 10πρ 6 p 2 dψe 5πρ ψe 6 p 2 3 = [γ − (1 + ) ] = [γ − (1 + ) ] (3.44) dV γWf 5 ρ dV γWVf (ρ + 3p) 5 ρ

Let α, β be constants. We deﬁne two alternative O.D.E. for the function u

du αρu − 2V (ρ + 3p) = 0 (3.45) dV du βu + ρu − 2V (ρ + 3p) = 0 (3.46) dV

If we integrate these two equations on an interval [a, b] using (ˆα, βˆ) = (α, 0) for (3.45) and (ˆα, βˆ) = (1, β) for (3.46) we get

1 1 b αρˆ (ν) + βˆ = exp( dν) (3.47) u(V ) u(b) ˆV 2ν(ρ(ν) = 3p(ν))

α is chosen in equation (3.45) so that u and ψe will match in a C1,1 way at V = b. This implies that

3p dlnψe d2ψe α = 2b(1 + ρ ) dV (b). In the case that limV →b+ dV 2 = 0, a simple calculation using d2ψe (3.33) shows that α = 1. If limV →b+ dV 2 > 0 a similar calculation shows that α < 1. d ln ψe In both cases, 0 < α ≤ 1, we have that 0 < 2V dV < α ≤ 1 at V = a. This fact is key in choosing Wf on certain intervals, which we will discuss more in detail later. If

39 d2ψe 1,1 limV →b+ dV 2 < 0, then an α that ensures a C match of u and ψe at b will be greater d ln ψe than 1. Therefore we lose the inequality 0 < 2V dV < 1 at a. This is precisely where d2ψe the equation (3.46) comes in. In the case that limV →b+ dV 2 < 0 we use (3.46). If we d ln ψe assume that ψe was chosen so that 0 < 2b dV (b) < 1 at V = b and choose β so that 0 < β < 3p(b), then at V = a (3.46) gives us

du du 0 = βu + ρ(a)u − 2V (ρ(a) + 3p(a)) =⇒ 2V (ρ(a) + 3p(a)) = βu + ρ(a)u dV dV (3.48)

(3.49)

Therefore

du 2V (ρ(a) + 3p(a)) = βu + ρ(a)u ≤ (3p(b) + ρ(a))u (3.50) dV d ln u (3p(b) + ρ(a)) =⇒ 2V ≤ < 1 (3.51) dV (ρ(a) + 3p(a))

where the last inequality follows from the fact that p increases into the star.

Now, if u satisﬁes equation (3.45), then we have an explicit expression involving

the scalar curvature Ru4g

4 2Wα p 2 u R 4 = (6(1 + ) − 5γ + γ(1 − α)) + 16πρ(1 − α) (3.52) u g 2 3p 2 ρ γV (1 + ρ )

6 p 2 In particular, note that if α = 1 and γ ≤ 5 (1 + ρ ) we have Ru4g ≥ 0. When u satisﬁes equation (3.46) we have an explicit expression involving the scalar curvature of the metric u4g given by

2 4 2W p 2 β 6β β u R 4 = (6(1 + ) (1 − ) − γ(5 + + )) − 16πβ (3.53) u g 2 3p 2 ρ 3p ρ ρ2 γV (1 + ρ )

40 3.6 The critical set and the oscillation set.

The function u described above will be used to define ψe on two types of intervals, the d2ψe intervals containing critical values of V and intervals where dV 2 oscillates indefinitely. We first describe the set of intervals which contain the critical values of V . We call this set U. The next lemma describes the construction of the sets in U.

Lemma 3.4 [ul-Alam (2007)]

Suppose V1 < Vs is not a critical value of V . Given any > 0 we can ensure that

critical values of V in [Vm,V1] are contained in a union of a ﬁnite number of disjoint Sk intervals [Vm, j0) ∪ ( n=1(in, jn)) = U such that the 1-dimensional Lebesgue measure of U, and 3-dimensional Hausdorﬀ measure of V −1(U) satisfy

max{L1(U),H3(V −1(U))} < (3.54)

Proof.

−1 Let > 0. We first want to show that the set C = {x ∈ V ([Vm,V1]) : W (x) = 0} has 3-dimensional Hausdorff measure 0. Suppose for contradiction that C has positive 3-dimensional Hausdorff measure. Then there exists an open 3-dim ball O(r) with

2 radius r which contains C and that O(r) \ C < 2 for any given 2 > 0. Therefore C 3 cannot have positive H measure. Next, for any [a, b] ⊂ [Vm,Vs) the co-area formula gives us [2]

b b 3 −1 −1 H ({x ∈ V ([a, b])|W (x) 6= 0}) = ( W 2 )dτ = f(τ)dτ (3.55) ˆa ˛V =τ,W 6=0 ˆa

−1 2 where we let V =τ,W 6=0 W = f(τ). This equality given by the co-area formula ¸ shows that the function f(τ) must be integrable on any interval [a, b] ⊂ [Vm,Vs) since 3 the H ([Vm,Vs)) < ∞. Using the continuity of integration we know that for our 1 given > 0 there exists a δ1 > 0 such that if S ⊂ [Vm,Vs) and L (S) < δ1 then

41 S f(τ)dτ < . The equality given by the co-area formula then tells us that ´

3 −1 −1 H ({x ∈ V (S)|W (x) 6= 0}) = ( W 2 )dτ = f(τ)dτ < (3.56) ˆS ˛V =τ,W 6=0 ˆS

1 Putting these things together we have that for any S ⊂ [Vm,Vs) such that L (S) < δ1 then

H3(V −1(S)) = H3((C ∩ V −1(S)) ∪ (Cc ∩ V −1(S))) (3.57)

= H3(C ∩ V −1(S)) + H3(Cc ∩ V −1(S)) (3.58)

= 0 + H3(Cc ∩ V −1(S)) (3.59)

< (3.60)

Our assumption that V is a C3 function on the complement of the 2-surfaces where ρ

has discontinuity gives us by Sard’s theorem [1] that the critical values of V in [Vm,V1] form a set of measure zero. Since this set has measure zero it can be contained in a countable union, denoted U, of disjoint open intervals such L1(U) < δ for any given

δ > 0. Since the interval (Vm,V1) is bounded we can arrange the open intervals so that Sk 1 U = [Vm, j0) ∪ ( n=1(in, jn)). If we choose δ < min(, δ1) then we have L (U) < δ < 1 3 −1 and L (U) < δ < δ1 implies H (V (U)) < .

Let U be the set given by Lemma 4 with respect to the V1 given by Lemma 3.2. For the set U we set out to make the spinor norm weighted scalar curvature integral over this set as small as we like. This is accomplished by choosing the function u described in the previous section. On intervals in U, we use equation (3.45) and 0 < α < 1

d2ψe if dV 2 > 0 at the right endpoint and we use equation (3.46) and 0 < β < 3p(b) if d2ψe 1,1 dV 2 < 0 at the right endpoint, and α or β is chosen so that u and ψe agree in C fashion at the right endpoint.

42 Let Cα,β be the constant given by

2W p C = max{ sup | α (6(1 + )2 − 5γ + γ(1 − α)) + 16πρ(1 − α)|, (3.61) α,β 2 3p 2 ρ γV (1 + ρ ) 2W p β 6β β2 sup | (6(1 + )2(1 − ) − γ(5 + + )) − 16πβ|} 2 3p 2 ρ 3p ρ ρ2 γV (1 + ρ )

−1 where the sup is taken over V ([Vm,V1]). This constant appears in the next lemma 2 2 which allows us to control the size of Ru4gu kξk on intervals in U. ´ Lemma 3.5 [ul-Alam (2007)]

Suppose that on [a, b] ⊂ (Vm,V1] u satisﬁes equation (3.45) or (3.46), with initial du dψe 1,1 conditions u(b) = ψe(b), and dV (b) = dV (b) where on [b, V1], ψe(V ) is C and 0 < d ln ψe 2V dV < 1. Suppose further that for V ≥ V1, ψe(V ) is as in Lemma 3.2. Then

2 2 −1 3 −1 | Ru4gu kξk dvolg| ≤ 4Cα,βVm H (V ([a, b])) (3.62) ˆV −1([a,b]) where the constant V1 is as in Lemma 3.2, and the constant Cα,β is as in (3.61).

Recall that on the set U there are only a ﬁnite number of intervals, i.e. U = Sk [Vm, j0) ∪ ( n=1(in, jn). Therefore, using Lemma 3.5 we can control the integral 2 2 Ru4gu kξk on all of U using the next lemma. On U we set ψe = u where u was ´ deﬁned by the construction above.

Lemma 3.6 [ul-Alam (2007)]

We have a constant C5 independent of and U from Lemma 3.4 such that for a ψe constructed above the total contribution in the spinor norm weighted scalar curvature integral from the set U is bounded by

2 2 2 2 | R 4 u kξk dvol | = | R ψ kξk dvol | < C (3.63) u g g ψe4g e g 5 ˆV −1(U) ˆV −1(U)

43 where is from Lemma 3.4. In case Vs is not a critical level set we can choose C5 to

be independent of the chose of V1

We want to deﬁne another set of intervals, U ⊂ [Vm,V1], and describe how we choose the conformal factor on U. The set U is deﬁned in order to capture intervals

d2ψe where dV 2 oscillates indefinitely. As in the set U, for U we define the function u which satisfy the DE described in the previous section. Let U ⊂ [Vm,V1] be a set of 6 p 2 intervals where |γ − 5 (1 + ρ ) | < 2 for some 2 > 0 which we will shortly define with motivation. Lemma 3.3 and the regularity conditions on the ρ(p) imply that intervals

d2ψe where dV 2 changes sign rapidly will be included in U.

d2ψe On intervals in U, we will have dV 2 = 0 at the right endpoint. We use equation (3.45) with α = 1 to define our function u, and the set ψe = u on intervals in U. The purpose of defining ψe = u on intervals in U is that we want to control the size of 2 2 V −1(U) Ru4gu kξk dvolg. To this end, and for motivating our definition of 2, we state ´ a lemma.

Lemma 3.7 [ul-Alam (2007)]

6 p 2 Suppose on [a, b] ⊂ [Vm,V1), |γ − 5 (1 + ρ ) |ψe(b) ≤ γ2, for some 2 > 0. Suppose d2ψe further that dV 2 (b) = 0. Then on [a, b] we can ﬁnd a positive function u with u(b) = du dψe ψe(b), dV (b) = dV (b) such that

2 2 −3 3 −1 | Ru4gu kξk dvolg| < 40πM2(Vmψe(b)) H (V ([a, b]))) ˆV −1([a,b])

Using Lemma 3.7 we get an integral bound on each interval in U. In order to extend this to all of U we need a global bound on (ψe(b))−2, i.e. a bound that holds for every right endpoint of intervals in U. This is given by following.

44 Lemma 3.8 [ul-Alam (2007)]

1,1 Suppose on [k, l], ψe is C . On some subintervals [k1, l1] of [k, l], ψe coincides with u

satisfying (3.45) or (3.46) with 0 < α ≤ 1 or 3p(l1) > β > 0 and on the rest of [k, l], 0 ≤ 2V d ln ψe ≤ 1. If ψe(l) > 0, then 1 ≤ l on [k, l]. dV (ψe(V ))2 V (ψe(l))2

In light of Lemma 3.2, we view Lemma 3.8 with interval [a, b] from Lemma 3.7

l Vs being a subinterval of [Vm,Vs]. We can then replace 2 with 2 . This gives V (ψe(l)) V (ψe(Vs)) us the following control on the integral from Lemma 3.7

2 2 −3 3 −1 | Ru4gu kξk dvolg| < 40πM2(Vmψe(b)) H (V ([a, b])) ˆV −1([a,b])

Vs 3 −1 ≤ 40πM2 H (V ([a, b])) (3.64) 3 2 Vmψe(Vs) ψe(b) 3/2 Vs 3 −1 ≤ 40πM2 H (V ([a, b]) (3.65) 9/2 3 Vm ψe(Vs) 3 −1 = 40πM2CVs,Vm H (V ([a, b])) (3.66)

3/2 where C = Vs . Lemma 3.7 classiﬁes what sets belong to U by choice of Vs,Vm 9/2 3 Vm ψe(Vs) 2. Setting 2 properly, we can use this inequality in order to mitigate the negative contribution on intervals in U. We know that the sum of H3(V −1([a, b])) over

3 −1 all possible intervals in U must be less than the total measure H (V ([Vm,Vs])). 3 −1 Defining 2 < MCVs,Vm H (V ([Vm,Vs]) and using this to define our set of intervals 2 2 U in Lemma 3.7 shows that we can make V −1(U) Ru4gu kξk dvolg as small as we like ´ when we properly define what constitutes intervals in the set U by specifying an 2

for Lemma 3.7. Therefore, we deﬁne U to be the set of intervals {[a, b] ⊂ [Vm,V1)} in d2ψe 6 p 2 3 −1 which dV 2 (b) = 0 and |γ − 5 (1 + ρ ) | < 2 where 2 < MCVs,Vm H (V ([Vm,Vs]))

45 3.7 The conformal factor on the regular set.

We have defined ψe on the set U and U. Our goal is to define ψe on [Vm,Vs]. We now describe the process for choosing Wf on (Vm,V1] \ U. This will aid us in defining ψe on the remaining parts of [Vm,Vs]. On these intervals Wf is used to define a ψe which satisfies equation (3.33). For a non-critical value τ ∈ (Vm,Vs] we define the function √ 2 −1 W kξk W (τ) = V (τ) (3.67) ave ¸ kξk2 √ V −1(τ) W ¸ Using the co-area formula we see that with this definition, and for the set U given in

Lemma 3.4 and V1 given in Lemma 3.2

2 Wave ◦ V − W 8d ψe 2 2 kξk dvolg = 0 (3.68) ˆ −1 3 dV V ((Vm,V1]\U) ψe

Using the fact that Wave(τ)−W (x) = Wave(τ)−Wf(x)+Wf(x)−W (x) and the above equation we have

2 2 Wf ◦ V − W 8d ψe 2 Wf ◦ V − Wave ◦ V 8d ψe 2 2 kξk = 2 kξk ˆ −1 3 dV ˆ −1 3 dV V ((Vm,V1]\U) ψe V ((Vm,V1]\U) ψe (3.69)

We want to now describe the method for defining Wf on sets in (Vm,V1] \ U. Our 2 Wf−Wave 8d ψe goal is to keep 2 ≥ 0. We define discontinuities in Wf in order to accomplish ψe3 dV this, as well as other goals. Specifically, suppose ψe has been defined on an interval [b, a] and we must give Wf a discontinuity at b. For the interval [c, b], Wf is defined so that d2ψe d2ψe either Wf < Wave if dV 2 (b) < 0 or so that Wf ≥ Wave if dV 2 (b) ≥ 0. For this purpose,

if Wf has been deﬁned on [b, a] and it happens that Wf − Wave changed signs at b and d2ψe dV 2 did not, then we give Wf a discontinuity at b. That is, we pick new initial data

r−(b), m−(b) for the ODE system, and compute Wf(b−) from those (slightly greater

or slightly smaller than Wf(b+), Wf = Wave ± δ). Then we solve the ODE system with the new data at b to ﬁnd, on some interval [c, b], r(V ), m(V ).We then use r(V ) and

46 d2ψe m(V ) to compute Wf(V ) on [c, b]. If dV 2 changes signs continuously at b and Wf−Wave d2ψe does not (so dV 2 (b) = 0), then we define ψe to be u on [c, b] where u satisfies equation d2ψe (3.45) with α = 1. Note that dV 2 (V ) = 0 persists on [c, b], while Ru4g ≥ 0 on this d2ψe interval. A big difference in this proof is that we allow intervals where dV 2 < 0. In previous work this was circumvented by placing restrictions on the equation of state.

d2ψe In this construction, when dV 2 < 0 we ensure that Wave − δ ≤ Wf < Wave, giving a discontinuity to Wf if needed. We also ensure that ψe remains C1,1 at b. In order to describe this process more rigorously, we start with a lemma.

Lemma 3.9 [ul-Alam (2007)] On any V -interval where m and r are positive solutions of equations (3.34)-(3.35), we

d ln ψe have 0 < 2V dV < 1.

d ln ψe The value of dV can be derived by using equation (3.37). In the case where m(b), r(b) are both positive, Lemma 3.9 is needed to ensure that we are able to deﬁne proper initial values at b when a discontinuity in Wf is needed. The next lemma guarantees we can change the initial data for (3.34),(3.35) at V = b, so as to produce the desired change in Wf, while preserving the condition given in Lemma 3.9.

Lemma 3.10 [ul-Alam (2007)] Suppose a set of solutions of equations (3.34)-(3.37) exists on [b, a] for some a and

d ln ψe 0 < 2V dV (b) < 1, ψe(b) > 0.

Suppose limV →b+ Wf := Wf+ = Wf(b) ≥ 0. Given δ > 0 we can ﬁnd positive

constants r−, and m−, Wf− such that Wf−,r−, and m− satisfy equation (3.36) at d ln ψe V = b, Wf+ + δ > Wf− > Wf+ and the value of dV computed from these constants using (3.37) remains the same at b.

47 Similarly, if Wf+ > 0, given δ > 0 we can ﬁnd positive constants r−m− and

Wf−,such that Wf−, r−, and m− satisfy equation (3.36) at V = b, Wf+ − δ < Wf− < Wf+, d ln ψe and the value of dV computed from these constants using (3.37) remains the same at b..

d ln ψe Lemma 3.10 requires 0 < 2V dV < 1 at b. By Lemma 3.9 this is true if m and r are both positive. In the case that m(b) = 0 but r(b) > 0 we have the following lemma.

Lemma 3.11 [ul-Alam (2007)] Suppose for initially positive solutions r(V ), m(V ), Wf(V ), ψe(V ) of equations (3.34)- (3.37) on [b, a], r(b) > 0, ψe is positive and C1,1 on [b, a]. Suppose further that m(b) = 0 d ln ψe 1,1 or V dV (b) = 0. Then we can ﬁnd a positive C conformal function ψe on the region

Vm ≤ V ≤ a for which the scalar curvature is nonnegative for Vm ≤ V ≤ b.

d2ψe Note that in the case that m(b) = 0 but r(b) > 0 we must have dV 2 > 0 just

before b. In this case we require Wf − Wave > 0. Since Wave is bounded away from zero, we know that Wf cannot equal zero at a point where m(b) = 0 unless r(b) = 0. This rules out the case of “irregular zeros” (deﬁned earlier). Gathering the previous lemmas we can now state the main lemma for giving needed discontinuities to Wf on

the set (Vm,V1] \ U.

Lemma 3.12 [ul-Alam (2007)] Suppose regular solutions r(V ), m(V ), Wf(V ), and ψe(V ) of equations (3.34)-(3.37)

exists on [b, a] ⊂ [Vm,Vs], r(V ), m(V ), Wf(V ), ψe(V ) are positive on [b, a] and ψe is 1,1 d ln ψe C on [b, a]. Suppose further that 0 < 2V dV (b) < 1. Then we can give a jump discontinuity to Wf at b (as small as desired, in either direction) so that ψe(V ) is a C1,1 d2ψe positive function on [d, b] for some d < b. When ρ is continuous at b the sign of dV 2

48 does not change at the discontinuity. When ρ is discontinuous at b, the limit from the

d2ψe left of limV →b− dV 2 has the same sign as it would be without the given discontinuity of Wf at b.

3.7.1 Accumulation of discontinuities.

There is an issue we need to address. There is a possibility that an inﬁnite number of discontinuities are needed in Wf and they accumulate at a point. If k is the number of discontinuities and they accumulate at a point b then it is possible that

d ln ψek limk→∞ 2V dV = 1. This would keep us from continuing to deﬁne Wf to the left of b using Lemma 3.12. It is also possible that in the case of an inﬁnite number of discontinuities we may not be able to control the sign of Wf − Wave no matter how small the initial value we give Wf. This issue is handled in two cases: when Wf is

increased above Wave to make Wf − Wave positive and when Wf is decreased below

Wave to make Wf − Wave negative. For the ﬁrst case we have the following lemma.

Lemma 3.13 [ul-Alam (2007)]

Suppose on [a, b] ⊂ (Vm,Vs] \ U that 0 < 2m < r and Wave + δ ≥ Wf ≥ Wave for some

δ > 0. If Wf(a) = Wave(a), Wf(b) = Wave(b) + δ, and c1 a Lipschitz constant of Wave δ on [a, b] then b − a > , where c2 = 8π(ρ(Vm) + p(Vm)) > 0. c2+c1

This lemma prevents the accumulation of intervals in (Vm,Vs] \ U such that Wf needs to be raised by placing a lower bound on the width of the interval on which the need to raise Wf can arise. However, note that this lower bound degenerates as δ → 0. Therefore, we can conclude that there can be at most a ﬁnite number of times

in [Vm,Vs] we will need to raise Wf above Wave. In the second case a similar result is not available. This follows from the fact that for the above lemma an upper bound

49 dWf on dV was available and utilized. For a similar result in the case that Wf is lowered dWf in order to make Wf < Wave we need a lower bound on dV which is not available. We must handle this case in a diﬀerent way. There is no way to guarantee that an accumulation of intervals in which Wf needs to be lowered will not happen. In the case an accumulation point occurs we show that the negative contribution to the spinor

norm weighted scalar curvature integral which occurs in (Vm,Vs] \ U can be oﬀset locally. This gives us non-negativity of the integral over an interval containing the accumulation point. The lemma also ensures that we can proceed further into the star using Lemma 3.12.

Lemma 3.14 [ul-Alam (2007)]

Let b ∈ [i, j] ⊂ (Vm,V1] \ U. Suppose we have a convergent sequence {bk}, bk ∈ [i, j]

such that b1 = b and on (bk+1, bk] we have rk(V ), mk(V ), Wfk(V ), and ψek(V ) are all 2 d ψek positive solutions to equations (3.34)-(3.37), dV 2 ≤ 0, Wfk < Wave, 0 < 2mk < rk, and at bk+1 (rk, mk, Wfk) is related to (rk+1, mk+1, Wfk+1) by the discontinuity constructed

in Lemma 10 with Wf+(bk+1) = Wave(bk+1) and Wf−(bk+1) = λk > 0. Denote limk→∞ bk by b∞. Then we can ﬁnd an N such that for the solutions (rk(V ), mk(V ), Wfk(V )) and k ≤ N we have

2 N−1 2 WfN − Wave d ψeN 2 X WfN − Wave d ψeN 2 2 kξk dvolg+ 2 kξk dvolg ≥ 0 ˆ −1 3 dV ˆ −1 3 dV V ([a,bN ]) ψeN k=1 V ([bk+1,bk]) ψeN

d ln ψeN for some a < b∞, and at a: 0 < 2V dV < 1.

Remark: In the ﬁrst integral we assume N is taken large enough, so that the solution

(rN , mN , WfN ) is deﬁned slightly to the right of b∞.

This result is precisely what we need in the case where an accumulation point of a sequence of discontinuities occur when trying to keep Wave − δ ≤ Wf < Wave in order to guarantee positivity of the integral on the interval and continue the construction

50 of Wf inward.

3.8 Convergence of a sequence of conformal factors.

Our goal on (Vm,V1]\U is to deﬁne Wf in such a way as to get a conformal factor which 2 Wave−W d ψe satisﬁes equation (3.33) with Wf replaced by Wave. This will give us Rb = 2 . ψe3 dV The spinor norm weighted scalar curvature integral over such an interval with this

scalar curvature will be zero (see (3.68)). This involves creating a sequence (Wfn) on

(Vm,V1] \ U and proving convergence for a subsequence of (ψen).

This process is as follows. Fix an interval [a, b] ⊂ (Vm,V1] \ U. In the above construction we started Wf at b Wf < Wave or Wf ≥ Wave, depending on the sign d2ψe of dV 2 . If Wf < Wave on [a, b] and |Wf − Wave| ≥ δ at some point in [a, b] then we introduce a discontinuity in Wf. At the left endpoint of intervals in U ∪ U we start d2ψe deﬁning Wf = Wave ±δ depending on the sign of dV 2 at the left endpoint of the interval d2ψe in U ∪ U. If Wf − Wave vanishes and changes signs while dV 2 remains the same sign, then we give a discontinuity to Wf. All of which uses Lemma 3.12. Notice that δ is d2ψe used to squeeze Wf closed to Wave when dV 2 < 0. To deﬁne the sequence we replace 1 δ above with n . In this case δ → 0 as n → ∞. Now, we denote the solution to equation (3.33) using Wfn on [a, b] as ψen. For each n,[a, b] is the union of two sets, 2 2 n n d ψen n d ψen S± where S+ denotes the set where dV 2 ≥ 0 and S− denotes the set where dV 2 < 0.

The construction of Wfn gives us that

2 Wfn − Wave 8d ψen 2 2 kξk ≥ 0 (3.70) ˆ −1 n 3 dV V ([a,b]∩S±) ψen

51 n Lemma 3.14 guarantees that the integral over S− is non-negative. The other is non- negative by construction. We now state a needed lemma.

Lemma 3.15 [ul-Alam (2007)] Suppose equation (3.70) holds and,

2 Wfn − Wave 8d ψen 2 2 kξk → 0 as n → ∞ (3.71) ˆ −1 3 dV V ([a,b]) ψen

Then b Wave dψen (1 − )(ρψen − 2V (ρ + 3p) dV ) | Wfn |dV → 0 as n → ∞ (3.72) ˆa Wave This lemma helps us satisfy a necessary hypothesis for our convergence lemma, which we state now.

Lemma 3.16 [ul-Alam (2007)] (n) (n) Suppose in an interval [a, b], the functions en, f1, f2, ψ1 and ψ2 satisfy

dψ(n) 1 = ψ(n) (3.73) ds 2 dψ(n) 2 = (1 − e )(f ψ(n) + f ψ(n)) (3.74) ds n 1 1 2 2

where f1 and f2 are piecewise continuous, and en are uniformly bounded measurable functions such that b (n) (n) (n) (0) a |en(f1ψ1 + f2ψ2 )|ds → 0 as n → ∞. Suppose further that ψi (a) = ψi (a) and ´ (n) |ψi (s)| are all bounded by a number independent of n. Then there is a subsequence (k) 1,1 {ψ1 } which converges to a C solutions ψ of the following equation with the initial (0) dψ (0) conditions ψ(a) = ψ1 (a) and ds = ψ2 (a),

d2ψ dψ = f ψ + f (3.75) ds2 1 2 ds

52 Lemma 3.16 is used to get the conformal factor ψe on [a, b] ⊂ (Vm,V1] \ (U ∪ U)

which satisﬁes equation (3.33) with Wf replaced by Wave. In using Lemma 3.16 in (n) conjunction with Lemma 3.15 we have ψ1 a solution of equation (3.33) with respect

2πρ 4πV (ρ+3p) Wave d d to Wfn, f1 = , f2 = , and en = 1 − . Also, = − which causes Wave Wave Wfn ds dV

the sign in front of f2 in equation (3.75) to be positive. The conclusion of Lemma 3.16 gives us our desired conformal factor ψe on [a, b]

We perform a similar scheme to intervals in U. We want ψe deﬁned on intervals in

U to satisfy equation (3.33) with Wf replaced with Wave. Recall that on intervals in U u is chosen to satisfy equation (3.45) with α = 1. We want to adapt Lemma 3.16

for this situation. Suppose [a, b] ⊂ U. Let [a, sn] be an interval such that s1 = b and

sn → a as n → ∞. Let χn be the characteristic function for the interval [a, sn]. For the function u chosen on intervals in U we have

d2u 1 6 p = [γ − (1 + )2]u (3.76) dV 2 3p 2 5 ρ 2V (1 + ρ )

(n) d2u (n) 2πρ Deﬁne f ψ = 2 and f = f + f χ where f is deﬁned as before, f = . 3 1 dV 1 1 3 n 1 1 Wave

4πV (ρ+3p) Wave We also deﬁne f2 as before, f2 = . en we deﬁne as before, en = 1 − , Wave Wfn except on [a, sn] where we set en = 0. We now have the following lemma for intervals in U.

Lemma 3.17 [ul-Alam (2007)] (n) The conclusion of Lemma 3.16 remains valid if we replace f1 by the function f1 .

This will give us the desired conformal factor ψe for intervals in U.

53 3.9 Conclusion of the proof.

Now, we put all of the lemmas above into our ﬁnal argument to prove the main

1 1 theorem. Let 0 < δn = n . First, we choose ψen(V ) for V ≥ Vs to be equal to 2 (1 + V ). We start with equation (3.42) for the spinor norm weighted scalar curvature integral.

By Lemma 3.2 we know there exists a noncritical value V1 < Vs, which does not

depend on n, and solutions (rn, mn, Wfn, ψen) of the spherically symmetric equations 1,1 (3.34)-(3.37) on [V1,Vs], with ψen > 0 which is C on [V1, 1) when adjoined to ψen

d ln ψen deﬁned above for [Vs, 1). Also, on [V1,Vs] we have 0 < 2V dV < 1, Wfn > Wave, and 2 d ψen dV 2 > 0. This gives us

2 Wfn − Wave d ψen 2 2 kξk > 0 (3.77) ˆ −1 3 dV V ([V1,Vs]) ψen

δn Let 0 < < be the chosen for Lemma 3.4 where C5 is the constant 2C5 that appears in Lemma 3.6. This gives us the set U which contains a ﬁnite number of intervals which contain the critical values of V . We can choose this

1 3 −1 to involve C5 to bound the measure of L (U) and H (V (U)) since the constant

C5 is independent of which and U is chosen in Lemma 3.4. We deﬁne U to be d2ψe 6 p 2 precisely those intervals {[a, b]} where dV 2 (b) = 0 and |γ − 5 (1 + ρ ) | < 2 where 3 −1 2 < MCVs,Vm H (V ([Vm,Vs])). Once we have the interval [V1,Vs] in place, we

progress inward into the star with initial conditions given at V1. Intervals will either

lie in U, U or (Vm,V1]\(U∪U). We will only have to deal with intervals by consequence of our regularity properties.

2 d ψen We start at V1 with dV 2 > 0. This means that apart from intervals in U, for 2 d ψen as long as dV 2 ≥ 0 we must have Wfn − Wave ≥ 0. If we come to a point b where 2 d ψen Wfn − Wave = 0 and changes signs to Wfn − Wave < 0 to the left of b but dV 2 remains

non-negative, we give Wfn a small discontinuity upwards (to make it larger than Wave) and start (rn, mnWfn) again at b, using Lemma 3.12.

54 2 d ψen If dV 2 = 0 at a point b and goes negative continuously to the left of b while 6 p 2 Wfn − Wave ≥ 0 then our action to the left of b depends on the sign of γ − 5 (1 + ρ ) 6 p 2 d3ψe at b. If γ − 5 (1 + ρ ) > 0 at b (note that this means the third derivative dV 3 (V ) > 0 at V = b) then we give a discontinuity to Wfn at b and set Wfn(b−) = Wave(b) − δn. If 6 p 2 γ − 5 (1 + ρ ) ≤ 0 at b then we use equation (3.45) and with α = 1 and deﬁne ψen = u 2 6 p 2 d ψen as long as γ − 5 (1 + ρ ) ≤ 0. The reason for the this is that if dV 2 has a zero and 3 d ψen then goes negative, then we want dV 3 > 0 for continuity. If this is not the case then 6 p 2 we want to redeﬁne ψen = u as an interval in U since a point where γ − 5 (1 + ρ ) = 0 2 d ψen will always be in an interval of U. Now, for as long as dV 2 < 0 we want to keep 2 d ψen Wave − δn ≤ Wfn < Wave. If this changes while dV 2 < 0 we introduce a discontinuity according to Lemma 3.12. Lemma 3.14 handles this case where the possibility of accumulation of discontinuity occurs by ensuring that the integral is positive on some

2 d ψen interval containing the accumulation point. We note that if dV 2 > 0, then by Lemma

3.13 there can be at most a ﬁnite number of times we will need to raise Wfn so that

Wfn − Wave > 0. At each of the points we need to raise Wfn we use Lemma 3.12. 1,1 At the right endpoint of these intervals we ensure that ψen remains C . At the left

d ln ψen endpoint of these intervals we maintain 0 < 2V dV < 1.

If we arrive at an interval in U then we deﬁne ψen = u with u satisfying equation 1,1 (3.45) with α = 1. α = 1 ensures that u connects to ψen in a C fashion at the right endpoint of the interval in U. At the left endpoint of this interval we have

d ln ψen 0 < 2V dV < 1. If we come to an interval [i, j] in U, then we set ψen = u and define 2 d ψen u on [i, j] according to equation (3.45) or (3.46), depending on the sign of dV at the 2 d ψen right endpoint, j. If limV →j+ dV > 0 then we use equation (3.45) with 0 < α < 1 2 d ψen to define u. If limV →j+ dV < 0 then we use equation (3.46) with 0 < β < 3p(j) to 2 d ψen define u. If limV →j+ dV = 0 then we use equation (3.45) with α = 1 to define u. At 2 d ψen i we start Wfn by setting, Wfn = Wave − δn if dV 2 < 0 at i or we set Wfn ≤= Wave + δn

55 2 d ψen 1,1 if dV 2 > 0 at i. The proper α, β chosen ensures u connects to ψen in a C fashion at

d ln ψen j. At the left endpoint of this interval, i, we have 0 < 2V dV < 1.

1,1 Suppose we have deﬁned Wfn and ψen up until the interval [b, 1] and that ψen is C d ln ψe on [b, 1), and 0 < 2V dV < 1 at b. We now want to continue to the next interval which has b as a right end point. The interval will be contained in one of three sets;

U,U, or (Vm,V1]\(U ∪U). Suppose the next interval is an open interval in U, say (a, b). The intervals in U are the intervals given by Lemma 3.4 which contain critical values

of V .(Note that for the sake of notation we use (a, b) where (a, b) = (in, jn) for some 2 d ψen n = 1..k.) There are only a ﬁnite number of these intervals. If limV →b+ dV 2 ≥ 0 then we deﬁne u to be a solution to equation (3.45) with 0 < α ≤ 1. This ensures that u(b)

2 1,1 d ln ψen d ψen agrees with ψen(b) in a C fashion and that 0 < 2V dV < 1 at a. If limV →b+ dV 2 < 0 then we deﬁne u to be a solution to equation (3.46) with 0 < β < 3p(b). This ensures

1,1 d ln ψen that u(b) agrees with ψen(b) in a C fashion and that 0 < 2V dV < 1 at a. In both cases, equation (3.47) implies that u is positive on (a, b). On this interval we set u(V ) = ψen(V ) for V ∈ (a, b) and continue inward. Sets in U are disjoint so we know 2 d ψen the next interval cannot be in U. Depending on the sign of dV 2 at a and which set the next interval lies in we progress inward. If the next interval [c, a] happens to be an interval in (Vm,V1] \ (U ∪ U) we start Wfn at a with Wfn = Wave ± δn depending on d2ψe the sign of dV 2 .

Now, suppose the interval [a, b] is an interval in (Vm,V1] \ (U ∪ U). In this case the endpoint a we do not ﬁx for this demonstration. The left endpoint a will be the

ﬁrst point in which we must alter Wfn. We start with Wfn at b according to Lemma 3.12, unless the previous interval was in U or U. If the previous interval was in U

2 d ψen or U then we deﬁne Wfn = Wave ± δn, where the ± is determined by the sign of dV 2 2 2 d ψen d ψen at b. If dV 2 > 0 at b then we choose Wfn so that Wfn = Wave + δn at b. If dV 2 < 0 2 d ψen at b then we choose Wfn so that Wfn = Wave − δn at b. If dV 2 = 0 and ρ, hence 2 2 d ψen d ψen dV 2 , is continuous at b, then on the interval [a, b] dV 2 will be positive or negative

56 6 p 2 depending on the equation of state using equation (3.44). If γ > 5 (1 + ρ ) at b then 2 2 d ψen 6 p 2 d ψen dV 2 < 0 on [a, b] and if γ < 5 (1 + ρ ) at b then dV 2 > 0 on [a, b] for some a < b. We must also address the issue of accumulation points. Suppose we have sequential

2 d ψen intervals Ij = [aj+1, aj] ⊂ (Vm,V1] \ (U ∪ U) where a1 = b, dV 2 ≥ 0, and at each aj a discontinuity is introduced to Wfn in order to raise Wfn above Wave at aj to maintain non-negativity of the integral over the interval. We know from lemma 3.13 that

M there can be at most a ﬁnite number of (Ij)j=1. Lemma 3.12 is used on each interval

Ij = [aj+1, aj]. Suppose we have sequential intervals Ji = [bi+1, bi] ⊂ (Vm,V1]\(U ∪U) 2 d ψen where b1 = b, dV 2 ≥ 0, and at each bi a discontinuity is introduced to Wfn in order

to maintain Wave − δn ≤ Wfn < Wave at bi which keeps the integral over the interval nonnegative. We know that the number of these intervals may tend to inﬁnity and an

accumulation of discontinuities (bi) can occur. Lemma 3.12 is used on each interval

Ji = [bi+1, bi]. In this case, Lemma 3.14 gives us an N such that for all i ≤ N we have

2 N−1 2 WfnN − Wave 8d ψenN 2 X Wfni − Wave 8d ψeni 2 2 kξk + 2 kξk ≥ 0 ˆV −1([a,b ]) ψ3 dV ˆV −1([b ,b ]) ψ3 dV N enN i=1 i+1 i eni (3.78) for some a < limi→∞ bi. This lemma ensures that the contribution of the integral is

d ln ψeN non-negative and at a 0 < 2V dV < 1 which allows us to continue construction

inwards of Wfn.

Finally, suppose that our interval [a, b] is contained in U with 0 < 2 < 3/2 δn in Lemma 3.7 where C = Vs . By deﬁnition this 80πC H3(V −1([V ,V ])) Vs,Vm 9/2 3 Vs,Vm m s Vm ψe(Vs) means that

6 p 2 |γ − (1 + ) |ψe(b) ≤ γ2 5 ρ

d2ψe and that dV 2 (b) = 0. In this case we deﬁne u to be a solution of equation (3.45) with

1,1 d ln ψen α = 1. α = 1 u connects to ψen at b in a C fashion and a we have 0 < 2V dV < 1.

57 On this interval we set u(V ) = ψen(V ) for V ∈ [a, b] and continue inward.

1,1 Now, we have deﬁned a conformal factor ψen which is globally C and uniformly bounded away from zero. We refer back to the expression in equation (3.42) and split the integral along the diﬀerent types of sets in order to address each type individually.

2 8 d ψen 2 0 = (Wfn − Wave(τ)) 2 kξk dvolg+ (3.79) ˆ −1 c 3 dV V (([Vm,V1]\U)∩U ) ψen X 2 2 Ru4gu kξk dvolg+ ˆ −1 α V (([Vm,V1]\U)∩U∩[iα,jα]) 2 8 d ψen 2 2 2 4 (Wfn − Wave(τ)) 2 kξk dvolg + Ru gu kξk dvolg ˆ −1 3 dV ˆ −1 V ([V1,Vs]) ψen V (U) + 4k∇ Θ k2dvol gb n gb ˆM

−1 c The integral over V (([Vm,V1] \ U) ∩ U ) when Wfn − Wave ≥ 0 is nonnegative by construction and when Wfn − Wave < 0 it is non-negative by Lemma 3.14. The

−1 −δn integral over V (U) we can make greater than 2 by Lemma 3.6. The integral over

−1 −δn V (([Vm,V1] \ U) ∩ U) is greater than 2 by Lemma 3.7 and the global bound from −1 Lemma 3.8. The integral over V ([V1,Vs]) is non-negative by Lemma 3.2. This gives us

2 8 d ψen 2 0 = (Wfn − Wave(τ)) 2 kξk dvolg+ (3.80) ˆ −1 c 3 dV V (([Vm,V1]\U)∩U ) ψen X 2 2 Ru4gu kξk dvolg+ ˆ −1 α V (([Vm,V1]\U)∩U∩[iα,jα]) 2 8 d ψen 2 (Wfn − Wave(τ)) 2 kξk dvolg+ ˆ −1 3 dV V ([V1,Vs]) ψen 2 2 2 R 4 u kξk dvol + 4k∇ Θ k dvol u g g gb n gb ˆV −1(U) ˆM

> −δn (3.81)

58 Now we want a conformal factor which satisﬁes equation (3.33) with Wave replacing

Wf on intervals in (Vm,Vs] \ U. Suppose [a, b] ⊂ (Vm,Vs] \ (U ∪ U). From equation (3.80) and (3.81) we see that

2 8 d ψen 2 0 ≤ (Wfn − Wave(τ)) 2 kξk dvolg (3.82) ˆ −1 3 dV V (([a,b]) ψen 2 8 d ψen 2 ≤ (Wfn − Wave(τ)) 2 kξk dvolg → 0 ˆ −1 c 3 dV V (([Vm,V1]\U)∩U ) ψen

and by Lemma 3.15 this gives us that

b Wave dψn (1 − )(ρψn − 2V (ρ + 3p) dV ) | Wfn |dV → 0 (3.83) ˆa Wave

as n → ∞. This allows us to use Lemma 3.16 with ψn = ψ , f = 2πρ , f = 4πV (ρ+3p) , 1 en 1 Wave 2 Wave

Wave and en = 1− . The conclusion of Lemma 3.16 give us the desired conformal factor Wfn ψe on [a, b].

Next, suppose [a, b] is an interval in U with the above construction. We again

want a conformal factor ψe deﬁned on [a, b] which satisﬁes equation (3.33) with Wave replacing Wf as discussed previously. Following the hypothesis given in Lemma 3.16,

on the interval [a, b] with χn the characteristic function on [a, sn] we deﬁne

Wave en = χn(1 − ) (3.84) Wfn 2πρ f1 = (3.85) Wave 4πV (ρ + 3p) f2 = (3.86) Wave d2u χ f u = χ (3.87) n 3 n dV 2 (n) f1 = f1 + χnf3 (3.88)

59 Then for each n, on [a, sn] we have

(n) ψ1 = u (3.89) dψ(n) du 1 = := ψ(n) (3.90) ds dV 2 dψ(n) d2u 2 = χ (3.91) ds dV 2 n d2u 2π du = 2 χn + (ρu − 2V (ρ + 3p) ) (3.92) dV Wave dV 2πρu d2u 4πV (ρ + 3p) du = + 2 χn − (3.93) Wave dV Wave dV du = f u + χ f u + f (3.94) 1 n 3 2 dV (n) (n) (n) = f1 ψ1 + f2ψ2 (3.95)

Also,

= 0 (3.98) where the last equality follows from the fact that u satisﬁes equation (3.45) with (n) α = 1. The above shows that on the interval [a, b] in U with f1 replacing f1 in Lemma 3.16, all of the hypothesis from Lemma 3.16 are met. Lemma 3.17 then gives us the desired conformal factor ψe on [a, b] in this case when [a, b] ⊂ U.

60 Finally, ψen(V ) for V ≥ Vs is ﬁxed for all n by equation (3.29). Rb on V ≥ Vs is −1 zero. This means for the integral over V ([Vs, 1)) we have

4k∇ Θ kdvol = 4k∇ Θk2dvol (3.99) gb n gb gb gb −1 −1 ˆV ([Vs,1) ˆV ([Vs,1)

−2 for all n. For V ≤ Vs on intervals in (Vm,Vs] \ U we have Θn = ψen ξ and Θ = −2 limn→∞ ψen ξ = ψξe where ψe is the limiting conformal factor. On compact intervals in

(Vm,Vs] \ U this convergence is uniform. On compact intervals in ((Vm,Vs] \ (U ∪ U)) the total integral is given by

2 8 d ψen 2 2 (Wfn − Wave) kξk dvolg + 4k∇gΘnk dvolg (3.100) ˆ 3 dV 2 ˆ b b ψen

Taking the limit as n → ∞ on these intervals gives us

2 8 d ψen 2 2 2 lim (Wfn − Wave) kξk dvolg + lim 4k∇gΘnk dvolg = 4k∇gΘk dvolg n→∞ ˆ 3 dV 2 n→∞ ˆ b b ˆ b b ψen (3.101) where the ﬁrst integral goes to zero by Lemma 3.16. For V ≤ Vs on intervals in U we have

2 2 2 R 4 u kξk dvol + 4k∇ Θ k dvol (3.102) nug g gb n gb ˆV −1(U) ˆM Taking the limit as n → ∞ on each of these intervals gives us

2 2 2 2 lim R 4 u kξk dvol + lim 4k∇ Θ k dvol = 4k∇ Θk dvol nug g gb n gb gb gb n→∞ ˆV −1(U) n→∞ ˆM ˆM (3.103)

where the ﬁrst integral goes zero by Lemma 3.17. This means, that on [Vm, 1) \ U

with ψe = limn→∞ ψen we have

0 = 4k∇ Θk2dvol (3.104) gb gb ˆM

61 Now, we know that 0 = 4k∇ Θk2dvol on [V , 1) \ U implies that ∇ Θ = 0 M gb gb m gb 2´ on [Vm, 1) \ U. Therefore ψe = kξk by our choice of ξ and continuity. We have a covariantly constant spinor ﬁeld Θ = ξ , with respect to the conformal metric g, kξk b deﬁned on [Vm, 1) \ U. We note here that our choice of ξ involved an arbitrary spinor

ξ0 which was constant at infinity. By definition of “constant at infinity” ξ0 is constant j with respect to an orthonormal frame ei near infinity where ei = ei ∂xj and (xj) are j asymptotically flat coordinates satisfying the mass decay conditions. ei being the “vielbein”. [17] We have a surjective map from the space of spinors to R3 given by

3 < vΘ, X >= Im < Θ,X · Θ > for X ∈ R (3.105)

3 where vΘ ∈ R . The fact that the map is surjective follows from the fact that Spin(3) is the double cover of SO(3). If Θ is a spinor ﬁeld generated by our construction then

−1 it is covariantly constant on V ([Vm, 1) \ U). This means that vΘ is also covariantly −1 constant on V ([Vm, 1)\U). Choose three spinors γi, i = 1..3 such that γi 7→ ei with the map above. (ei) is an orthonormal frame of TpM at some point p near inﬁnity.

Θi = γi, i = 1..3 are covariantly constant with respect to the conformal metric gb −1 on V ([Vm, 1) \ U). Take a diagonal sequence in the conformal factors converging to Θi. This diagonal sequence provides a conformal factor which makes all three spinors covariantly constant. This implies that all three vectors ei are covariantly constant. In particular, this gives us a that the frame (ei) is covariantly constant −1 −1 on V ([Vm, 1) \ U). We can extend the constant frame ﬁeld (ei) to the set V (U) by continuity. Therefore, we have a constant frame for TM, globally. Hence the conformal metric is ﬂat.

We recall the tensor used to prove spherical symmetry from conformal ﬂatness [22]

4 abc 2 2 2 V RabcR = 8W kΩk + k∇T W k (3.106)

62 In the case that the conformal metric is ﬂat, this tensor vanishes and standard arguments imply that the original metric has to be spherical. In the vacuum, V

is a connected level set. When the above tensor vanishes in the vacuum, V ≥ Vs, we see that W must be constant, which follows from the second term vanishing in equation (3.106). We assume without loss of generality that Vs has only one connected component. This implies that Vs is a noncritical level set of W . Lemma 3.6 in this case implies that the constant C5 is independent of V1.

We started with a static stellar model with equation of state piecewise C1. We were able to construct a conformal factor ψ which forced ∇ Θ = 0 everywhere outside e ψe4g of sets which contain critical values of V . This allows us to define an orthonormal frame field that is covariantly constant everywhere outside of U. We extend the frame field to U which gives us a global, constant frame field on TM. We conclude that the conformal metric is flat. Standard arguments using the vanishing of the tensor equation in (3.106) in the case of a flat conformal metric implies that our original metric g must be spherically symmetric. This concludes the proof that given a static stellar model it must be spherically symmetric.

63 Chapter 4

Physical Constraints on Stellar Models

The mathematical framework of General Relativity provides physical constraints to stellar models. Assuming the validity of the theory of general relativity, we can exclude certain physical phenomena from existing. We may also draw conclusions regarding the physically unlikely. In this section we derive a lower bound on ∆ = 2Gm(R) 1 − c2R in the case of a finite, static, spherically symmetric stellar model in which the density is a monotonically decreasing function of r. In the Newtonian limit c → ∞, ∆1/2 corresponds to the Newtonian gravitational potential. The lower bound on ∆ gives us bounds on the value of the metric potential at the boundary, as well as an upper bound on the mass in the case of a fixed radius R, known as “Buchdahl’s inequality”. We also explore a constraint on the adiabatic index of a finite, static stellar model which is not necessarily spherically symmetric. This is directly related to the proof of uniqueness for a static star. We show that the constraint on the adiabatic index offers support to the “Fluid Ball Conjecture”. Finally, we look at necessary and sufficient conditions ensuring the finiteness of the radius of a static, spherically symmetric stellar model. This section follows work from Buchdahl [9], Lindblom and ul-Alam [33], and Rendall and Schmidt [7].

64 4.1 Buchdahl’s Inequality

Suppose we have a ﬁnite, static, spherically symmetric stellar Model with a perfect ﬂuid source, a radius R and mass m(R). The metric for such a stellar model is given by

ds2 = −c2ea(r)dt2 + eb(r)dr2 + r2(dθ2 + sin2θdφ2) (4.1)

In this section we will use many of the expressions derived in chapter 2. The motivation of these expressions follow from the work of Buchdahl [9]. We now recall some of these expressions. We consider the system derived in equations (2.5)-(2.7). We also consider the auxiliary equations given in (2.16) and (2.33).

r m(r) = 4π s2ρ(s)ds (4.2) ˆ0 m(r) w(r) = (4.3) r3

We also recall expression (2.46)-(2.48).

2G y2(r) = e−b(r) = 1 − r2w(r) (4.4) c2 a(r) z(r) = e 2 (4.5)

x(r) = r2 (4.6)

Finally, recall the expression from (2.64).

d dz G 1 dw (y ) − ( )z = 0 (4.7) dx dx 2c2 y dx

65 Theorem 4.1 [Buchdahl (1959)] Given a finite, static, spherically symmetric perfect fluid stellar model of radius R and mass m(R) such that the density of the perfect fluid does not increase outward, dρ i.e. dr ≤ 0 we have 1 ∆ ≥ 9

2G m(R) where ∆ = 1 − c2 R

Proof. We ﬁrst consider the expression in (4.7) for r ≤ R and do a change of variable. Let

x 1 ξ(x) = dx (4.8) ˆ0 y(s) This way, we can expression (4.7) as

d dz ( ) − g(ξ)z (4.9) dξ dξ

G 1 dw where g(ξ) is equal to 2c2 y dx in terms of ξ. Since we assume that the density of the star does not increase outward it follows that

dw dw dξ dx dw 1 0 ≥ = = 2r (4.10) dr dξ dx dr dξ y(x)

dw which implies that dξ ≤ 0. Therefore g(ξ) ≤ 0 since y ≥ 0. Since g(ξ) ≤ 0 we have the inequality

d dz d dz 0 = ( ) − g(ξ)z ≥ ( ) (4.11) dξ dξ dξ dξ This inequality, of course, implies that

dz dz dz | ≤ ≤ | (4.12) dξ r=R dξ dξ r=0

dz We can need an expression for dξ . Recall equation (4.5).

66 dz dz dr dx = (4.13) dξ dr dx dξ dz y(r) = (4.14) dr 2r

dz Next, we want to evaluate dξ at the boundary of our star, i.e. r = R. Recall Birkhoﬀ’s theorem [10]. Continuity implies that at the boundary our interior solution must match the exterior solution, Schwarzschild’s vacuum solution in this case. So

a(R) 2G m(R) 2 a(R) y(R) = e and e = 1 − c2 R . Therefore, we have

dz dz y(R) | = (4.15) dξ r=R dr 2R a0(R) 1 = ea(R) (4.16) 2 2R 2G m(R) 1 = [1 − ] a0(R) (4.17) c2 R 4R 2G m(R) 1 1 2G m(R) = [1 − ] [ ] (4.18) c2 R 4R 2G m(R) c2 R2 [1 − c2 R ] G m(R) = (4.19) c2 2R3 G 1 = w(R) (4.20) c2 2

Going back to the inequality in (4.12) and using (4.20), we get

G 1 dz dz dz dr dx dz y w(R) = | ≤ = = (4.21) c2 2 dξ r=R dξ dr dx dξ dr 2r G r dz w(R) ≤ (4.22) c2 y dr If we integrate (4.22) with respect to r from r = 0 to r = R we get

R G r 1 r=R 2 2 w(R) dr ≤ z(r)|r=0 = ∆ − z(0) (4.23) c ˆ0 y

67 Recall the deﬁnition of y2 from (4.4) and the fact that w0 ≤ 0. This means that

w(R) ≤ w(r) (4.24) and

2G 2G y2(r) = 1 − r2w(r) ≤ 1 − r2w(R) (4.25) c2 c2 which implies that

2G 2 1 y(r) ≤ [1 − r w(R)] 2 (4.26) c2

2Gw(R) Let Ce = c2 . Using (4.26) in (4.23) gives us

R 1 r ∆ 2 − z(0) ≥ w(R) dr (4.27) ˆ0 y R r ≥ w(R) dr (4.28) 2G 1 ˆ 2 2 0 [1 − c2 r w(R)] 2 w(R) 1−CRe 1 = − 1 du (4.29) 2Ce ˆ1 u 2 w(R) 2 1 = 1 − (1 − CRe ) 2 (4.30) Ce c2 = [1 − y(R)] (4.31) 2G 2 c 1 = [1 − ∆ 2 ] (4.32) 2G

Note that z(0) ≥ 0 and therefore gives us the inequality

2 1 c 1 ∆ 2 ≥ [1 − ∆ 2 ] (4.33) 2G 2 2G 1 c 1 ∆ 2 + ∆ 2 ≥ 1 (4.34) c2 c2

68 2 1 c 1 ∆ 2 ≥ ≥ (4.35) 2G + c2 3 Finally, we see that

1 ∆ ≥ (4.36) 9 As a corollary to Theorem 4.1, we can just note the deﬁnition of ∆ and algebraically manipulate equation (4.36) to ﬁnd that

m(R) c2 4 ≤ (4.37) R G 9

c2 4 m(R) ≤ R (4.38) G 9 Therefore, we have an upper bound on the mass of a finite, static, spherically symmetric stellar model given a fixed radius R. As a side note, the derivation of the upper bound on the mass in the case of a fixed radius is completely independent of pressure.

If we assume that we have a ﬁnite, static, spherically symmetric stellar model but parametrized by the potential V so that the line element takes the form

2 2 2 i j ds = −V dt + gijdx dx (4.39)

for i, j = 1, 2, 3 and V , gij only depends on r we can make another observation. Let

V (R) = Vs denote the potential at the surface of the star. By continuity of the metric and Birkhoﬀ’s theorem we know that

1 V 2 = ∆ ≥ (4.40) s 9 Hence, we have a lower bound on the surface potential

69 1 V ≥ (4.41) s 3

4.2 Constraints on the Adiabatic Index

Suppose now that we have a ﬁnite, static stellar model that is not necessarily spherically symmetric with a perfect ﬂuid matter model. In this case we have the metric given by

2 2 2 i j ds = −V dt + gijdx dx (4.42) for i, j = 1, 2, 3 and V and gab are time-independent. We assume that we also have an equation of state, p(ρ). We denote the adiabatic index of our ﬂuid by γ and deﬁne it by

ρ + p dp γ(p) := (4.43) p dρ We now state a theorem. Theorem 4.2 [Lindblom and Masood-ul-Alam (1993)] Consider a static stellar model in general relativity theory whose surface occurs at a ﬁnite radius.(This assumes asymptotic conditions on V and gij) Assume that the equation of state ρ = ρ(p) is a positive and non-decreasing C1 function of the pressure.

1 + Assume that γ(p) is bounded as p → 0 . Then the adiabatic index must satisfy the inequality 6 p 6 γ > (1 + )2 ≥ 5 ρ 5 at some point within the star.

70 We suppose that our static stellar model has a surface potential of 0 < Vs < 1. Recall from chapter 3 expressions (3.16), and (3.17)-(3.18). We can express the pressure and density in terms of the potential,V .

p 1 V h(p) := ds = ln( s ) (4.44) ˆ0 ρ(s) + s V V p(V ) = h−1[ln( s )] (4.45) V ρ(V ) = ρ(p(V )) (4.46)

The proof for theorem 4.2 follows the technique we used to derive spherical symmetry in chapter 3. We will deﬁne a conformal factor which transforms the spatial metric gij and then make an observation about the adiabatic index in the case that the spatial metric is conformally ﬂat.

First, we deﬁne our conformal factor. The conformal factor for V ≥ Vs is deﬁned to be

1 ψ(V ) = (1 + V ) (4.47) 2 This conformal factor showed up in chapter 3 during our proof of uniqueness of the static stellar model. The conformal factor we use for 0 ≤ V ≤ Vs is given by the expression [33]

Vs 1 Vs ρ(s) ψ(V ) = (1 + Vs) exp − ds (4.48) 2 1 + Vs ˆV s[ρ(s) + 3p(s)]

First note that (4.47) and (4.48) agree at the boundary, VS. To prove continuity of the ﬁrst derivative of the conformal factor we need a lemma.

71 Lemma 4.1 [Lindblom and Masood-ul-Alam (1993)] Consider a static stellar model in general relativity theory whose surface occurs at a ﬁnite radius. Assume that the equation of state ρ = ρ(p) is a positive and non- decreasing function in some open neighborhood of the surface of the star: i.e. for

+ p pressures 0 < p < . Then lim p → 0 ρ = 0.

Proof. Recall equation (4.44). Since the equation of state ρ(p) is a non-decreasing function we can estimate the integral.

p 1 h(p) = ds (4.49) ˆ0 ρ(s) + s p 1 ≥ ds (4.50) ˆ0 ρ(p) + s p = ln(1 + ) (4.51) ρ

We know that limp→0+ h(p) = 0, so raising (4.49) and (4.51) using exp and taking the limit we get

p 0 = lim eh(p) − 1 ≥ lim (4.52) p→0+ p→0+ ρ since p, ρ ≥ 0 we see that p lim = 0 (4.53) p→0+ ρ Proving Lemma 4.1.

Now, we want to show that the ﬁrst derivative of ψ is continuous at VS. Computing the derivative of ψ in the interior of the star gives us [33]

dψ V ψ(V ) = s (4.54) dV 3p V (1 + Vs)(1 + ρ )

72 Recall that Vs is deﬁned to be where p(V ) = 0. So taking the limit of (4.54) as

V → Vs and recalling the lemma gives us

dψ Vsψ(V ) |Vs = lim (4.55) dV − 3p V →Vs V (1 + Vs)(1 + ρ ) V ψ(V ) = s s (4.56) Vs(1 + Vs) 1 = (4.57) 2

dψ Comparing this to the derivative of ψ for V ≥ Vs deﬁned in (4.47) reveals that dV is

also continuous at the boundary. The second derivative of ψ vanished for V ≥ VS.

For V ≤ Vs we have[33]

d2ψ V ψ(V ) 2 + 3V 3 p = s s − (1 + )2 (4.58) dV 2 3p 1 + V γ ρ V (1 + Vs)(1 + ρ ) s

d2ψ We can see from this expression that dV 2 is continuous on the interior of the star 1 and is bounded since we assume that γ is bounded. Therefore the ﬁrst derivative is Lipshitz. Hence the conformal factor we have deﬁned ψ is C1,1.

Proof of Theorem 4.2.

4 We deﬁne our conformal metric to be geij = ψ (V )gij. Our choice of conformal metric for V ≥ Vs ensures that the mass of geij equals zero. The expression for the conformal scalar curvature Re is given by the following expression.[33]

5 8 i Re = (2πρψ − D Diψ) (4.59) ψ where Di is the covariant derivative with respect to the spatial metric gij. Using the expression in (3.13) and expressions (4.47)-(4.48), (4.54), and (4.58) we can derive an expression for the conformal scalar curvature. The result is given by the following.[33]

73 2 i 4 8ρ VsD VDiV p 2 2 + 3Vs ψ (1 + Vs)Re = 16πρ(1 − Vs) + 2 2 3(1 + ) − γ (4.60) γV (ρ + 3p) ρ 1 + Vs

This expression holds on the interior of the star. Our chose of conformal factor for

V ≥ Vs gives us that Re = 0 for V ≥ Vs. Recall that for any stellar model that

has ﬁnite radius the potential on the surface, VS is strictly less that one [33]. The expression in (4.60) reveals that the conformal scalar curvature Re is non-negative as long as

p 2 + 3V 3(1 + )2 − γ s ≥ 0 (4.61) ρ 1 + Vs or equivalently, p 1 + V γ ≤ 3(1 + )2 s (4.62) ρ 2 + 3Vs We recall the Positive Mass Theorem discussed in chapter 3. We note that with

our deﬁned conformal factor ψ the conformal metric geij and asymptotic conditions

assumed for gij allows us to apply the rigidity statement of the Postive Mass Theorem if Re is nonnegative. Re is non-negative precisely when the adiabatic index satisﬁes the inequality in (4.62). If the inequality in (4.62) is satisﬁed, then the Positive Mass

Theorem implies that geij is ﬂat. In this case Re must vanish. If Re = 0 then by equation

(4.60) we see from the ﬁrst term on the right that Vs = 1 and in the second term on the right that 3(1 + p )2 = γ 2+3Vs , which implies equality in (4.62). However, this ρ 1+Vs

contradicts the fact that we have a ﬁnite stellar model since Vs must be strictly less than 1 if the star is ﬁnite. Therefore, we must assume that Re cannot be everywhere non-negative. Hence, there must be at lease one point where Re < 0. At this point, (4.62) is violated. Hence, at the point where Re < 0 we must have the inequality

p 1 + V γ > 3(1 + )2 s (4.63) ρ 2 + 3Vs

74 Futhermore, noting that Vs < 1 in the case of a ﬁnite star, we have

p 1 + V γ > 3(1 + )2 s (4.64) ρ 2 + 3Vs 6 p ≥ (1 + )2 (4.65) 5 ρ 6 ≥ (4.66) 5

This concludes the proof of Theorem 4.2.

As a corollary to this theorem, we consider the case where our star has a polytropic equation of state, i.e. an equation of state of the form

1+ 1 p(ρ) = κρ n (4.67)

where κ and n are constants. The constant n is known as the “polytropic index”. The constraint on the adiabatic index γ in equation (4.65) gives us a constraint on the polytropic index.

5 n < 6p ≤ 5 (4.68) 1 + ρ Therefore, this oﬀers a direct analog to the Newtonian limit on the polytropic index, n < 5 [34].

We make one further observation in this section. Recall the expression K deﬁned by Beig and Simon in [35] and recall “Condition B” from Lindblom and Masood-ul- Alam [29]. “Condition B” was needed in the proof of spherical symmetry of static stellar models. In fact, “Condition A” in [29] implies the ﬁrst of the inequalities of “Condition B”. “Condition B” is given by the inequality [29]

5ρ2 10V 2 ≥ κ > s (4.69) 6p(ρ + 3p) exp[2h(p)] − 1

75 ρ+p dρ where κ := ρ+3p dp . “Condition B” implies the upper bound on κ. This upper bound on κ is equivalent to K ≤ 0 given by Beig and Simon[35]. If we analyze the upper bound on κ we see that

5ρ2 ρ + p dρ ≥ κ = (4.70) 6p(ρ + 3p) ρ + 3p dp 5 ρ + p dρ p(ρ + 3p) ρ + p dρ ≥ = p (4.71) 6 ρ + 3p dp ρ2 ρ2 dp 6 ρ2 1 dp ρ2 + 2ρp + p2 1 dp 1 dp ≤ < = (ρ + p) = (4.72) 5 ρ + p p dρ ρ + p p dρ p dρ 6 ρ + p dp < = γ (4.73) 5 p dρ So we see here that Buchdahl’s inequality manifest itself in the condition necessary for spherical symmetry. Therefore, Buchdahl’s inequality rules out the existence of a large class of potential counter examples to the Fluid Ball Conjecture.

4.3 Necessary and Suﬃcient Conditions for a Finite Radius

In chapter 2 we proved the existence of spherically symmetric, static stellar models with perfect fluid source. These model stars were either finite or infinite. In the case the model star was finite the pressure became zero at an R < ∞ and the exterior Schwarzschild solution matched the interior solution at the boundary. In the case the model stars was infinite the density ρ > 0 for all r ≥ 0 and limr→∞ ρ(r) = 0. This section looks at criteria based on the equation of state which supplies sufficient and necessary conditions for a star to posses a finite radius.

In the case that ρ → 0 as r → ∞, if p(R) = 0 for some R < ∞ and ρ(R) > 0 then the star is obviously finite. So consider the case when p and ρ vanish simultaneously. Our first goal is to derive a sufficient condition for a spherically symmetric, static

76 stellar model to have a ﬁnite radius. We recall several equations from chapter 2: (2.16), (2.33), and the T.O.V. equation in (2.36).

r m(r) = 4π s2ρ(s)ds (4.74) ˆ0 m(r) w(r) = (4.75) r3 dp 2G −1 4πp p = −Gr 1 − r2w ( + w)( + ρ) (4.76) dr c2 c2 c2

Theorem 4.3 [Rendall and Schmidt(1991)] dρ Consider a spherically symmetric, static stellar model such that dr < 0,the equation ∞ of state is C , and has a central pressure given by p0. Then a suﬃcient condition for a ﬁnite radius is p0 1 2 < ∞ ˆ0 (ρ(s))

Proof. dρ Since we assume that dr < 0 then equation (4.74) gives us the inequality

r m(r) = 4π s2ρ(s)ds (4.77) ˆ0 r ≥ 4π s2ρ(r)ds (4.78) ˆ0 4 = πρ(r)r3 (4.79) 3

If we take this inequality and recall (4.75), this gives us another inequality

4 m(r) 4 m(r) ≥ πρ(r)r3 =⇒ w(r) = ≥ πρ(r) (4.80) 3 r3 3 Using the T.O.V. equation in (4.76) and a change of variable x = r2, in conjunction with the inequality in (4.80) we get

77 dp dx 2G −1 4πp p = −Gr 1 − r2w ( + w)( + ρ) (4.81) dx dr c2 c2 c2 < −Grwρ (4.82) 4 4Gπ ≤ −Grρ πρ = − rρ2 (4.83) 3 3

dx Since dr = 2r, the above inequality gives us

dp 2Gπ < − ρ2 (4.84) dx 3 We now deﬁne p(x) to be the solution of

dp 2Gπ = − ρ2 (4.85) dx 3

given the same central pressure p0 and the same equation of state, ρ(p). This gives us the inequality

p(x) < p(x) (4.86)

The diﬀerential equation for p given in (4.85) can easily be solved.

p(x) 1 2Gπ 2 ds = − x (4.87) ˆp0 (ρ(s)) 3

Now, if p(x1) = 0 for some x1 < ∞, then there must be some x2 ≤ x1 such that p(x2) = 0 by equation (4.86). But, if the integral

p0 1 2 ds < ∞ (4.88) ˆ0 (ρ(s)) then there must be an x1 < ∞ where p(x1) = 0. Therefore, for some x2 ≤ x1 < ∞ we have p(x2) = 0. Hence, the star is ﬁnite.

78 We now look at a necessary condition for a spherically symmetric, static star to be ﬁnite.

Theorem 4.4 [Rendall and Schmidt(1991)] dρ Consider a spherically symmetric, static stellar model such that dr < 0,the equation ∞ of state is C , and has a central pressure given by p0. Then a necessary condition for a ﬁnite radius is p0 1 −2 ds < ∞ ˆ0 c s + ρ(s)

Proof. Suppose we have a solution to the spherically symmetric stellar model guaranteed

by Theorem 2.2 from chapter 2 with a central pressure given by p0. Suppose also 2 that the boundary of this star occurs at a ﬁnite radius x = x1 where x = r . So

by deﬁnition the pressure p(x1) = 0. Consider the compact interval [0, x1] and the continuous quantity

2G 1 − xw (4.89) c2

This continuous quantity attains a minimum as some x0. Let this minimum value be denoted A. Looking at the quantity it is not hard to see that A ≥ 0. Recalling once more the T.O.V. equation given in this chapter by (4.76), but in terms of x, we have the inequality

dp G 2G −1 4πp p = − 1 − xw ( + w)( + ρ) (4.90) dx 2 c2 c2 c2 1 4πp p > −G ( + w)( + ρ) (4.91) A c2 c2 p = −B( + ρ) (4.92) c2

G 4πp0 where B = − A ( c2 + w(0)). Similar to the proof of Theorem 4.3, we deﬁne p to be the solution of

79 dp p = −B( + ρ) (4.93) dx c2

with a central pressure given by p0 and the same equation of state. We can easily solve equation (4.93).

p 1 −2 = −Bx (4.94) ˆp0 c s + ρ(s)

In this case, we see that p(x) < p(x). Since we assume that p(x1) = 0, then p(x1) = 0 since pressure is non-negative. This implies that

p0 1 −2 = Bx1 < ∞ (4.95) ˆ0 c s + ρ(s) This completes the proof.

80 Bibliography

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85 Appendix

86 Appendix A

Spin Geometry

We want to give a brief overview of spin geometry and use it to derive Witten’s expression for the ADM mass of an asymptotically ﬂat Riemannian manifold. This appendix follows closely to Friedrich’s book ”Dirac Operators in Riemannian Geometry”. We begin with the construction of spinors in the setting of Minkowski space. We then generalize this construction to the setting of bundles where we describe the connection and curvature for the spinor bundle. Lastly, we derive the formula of Lichnerowicz and show how Witten used this in his construction of the formula for mass in his proof of the Positive Mass Theorem.

A.1 Dirac Spinors

Let (M 4, η) denote Minkowski space. Our ﬁrst goal is the construction of the Dirac Spinors. To this end, we start with the construction of the Cliﬀord algebra. The

Cliﬀord algebra for (M 4, η) is an associative algebra over R with unit and a linear map c : M 4 → CL(M 4) which satisﬁes the relationship

c(x).c(y) + c(y)c(x) = −2 < x, y >η .Id (A.1)

87 4 3 where the “low dot” represents Cliﬀord Multiplication and x, y ∈ M .[14]. Let {ei}i=0 be an orthonormal basis for M 4. The Cliﬀord algebra contains the relation

c(ei).c(ej) + c(ej).c(ei) = 0 for i 6= j (A.2)

In this case, we also get a basis for the 24 = 16 dimensional vector space CL(M 4).

4 This basis of CL(M ) is formed by the elements Id and ei1 .ei2 . ··· eis where 1 ≤ i1 < i2 < ··· < is ≤ n with 1 ≤ s ≤ n.[14]

Our goal is the construction of the Dirac spinors. The Dirac spinors, which we denote S, is a 4 dimensional complex vector space which acts as a representation space for an irreducible representation of a complexifed Clifford algebra. There are two metric signatures that are common when describing Minkowski space. They are (−, +, +, +), which is common in math literature, and (+, −, −, −) which is common in physics literature. In the setting of spinors it is more natural to work with the signature (+, −, −, −). Each one of these metric signatures will generate a Clifford algebra. These Clifford algebras are not the same. In our case it is not necessary to prescribe a metric signature for η since we are constructing a complexified Clifford algebra. This is no loss of generality since

4 4 CL4 ≡ CL(M )1,3 ⊗R C ≡ CL(M )3,1 ⊗R C (A.3)

In other words, the complexiﬁcation of the Cliﬀord algebras generated by both metric signatures will be isomorphic.[13]

The Cliﬀord algebra CL(M 4) has a representation space of a 4 dimensional complex vector space. We denote κ to be the so-called Spin representation of the

Cliﬀord algebra CL(M 4). In other words, our representation is simply the following isomorphism.[14]

88 4 κ : CL(M ) → End(S) (A.4)

We can now define the notion of Clifford multiplication of vectors and spinors. Let v ∈ M 4. Under the representation, κ(c(v)) is an endomorphism of the space of Dirac spinors, S. We define a linear map

4 µ : c(M ) ⊗R S → S (A.5)

For v ∈ M 4 and ψ ∈ S we have

µ(c(v) ⊗ ψ) = κ(c(v))ψ = c(v).ψ (A.6)

We now want to describe a certain subgroup of the Cliﬀord algebra, Spin(4) ⊂

CL(M 4). Consider the sub-group of CL(M 4), the non-complexiﬁed Cliﬀord algebra generated by (M 4, η), which is multiplicatively generated by {c(v)|v ∈ M 4 and kvk = 1}. We call this group P in(4) ⊂ CL(M 4). The elements in P in(4) are the products

4 c(v1).c(v2) ··· .c(vm) with vi ∈ M and kvk = 1. Let O(4) denote the group of orthogonal transformations of M 4. There exists a continuous surjective group homomorphism from P in(4) onto the group O(4).[14] We denote this map

λ : P in(4) → O(4) (A.7)

In order to define this map we must first define another map. Every Clifford algebra carries an anti-involution denoted

γ : CL(M 4) → CL(M 4) (A.8)

such that

c(v) = γ(c(v))for v ∈ M 4 (A.9)

89 γ possess several properties.[14]

1.) γ is Linear. 2.) γ ◦ γ = Id 3.) γ(c(v)) = c(v) for all v ∈ M 4. 4.) γ(x.y) = γ(y).γ(x) for x, y ∈ CL(M 4).

The homomorphism λ : P in(4) → O(4) is then given by

λ(x): M 4 → M 4, λ(x)v = x.c(v).γ(x) (A.10)

for x ∈ P in(4) ⊂ CL(M 4) and v ∈ M 4. Now, let SO(4) denote the orthogonal transformations of M 4 which have unit determinant. Then the group Spin(4) ⊂ CL(M 4) is given by

Spin(4) = λ−1(SO(4)) (A.11)

and ker(λ) = {−1, 1} ≡ Z2. We also state the fact that Spin(4) is simply connected. Therefore Spin(4) is the universal double cover of the group SO(4) and λ is the

covering map. Now, for the complexiﬁed Cliﬀord algebra CL(M 4) the group Spin(4)C is given by the following.

C 1 Spin(4) = Spin(4) ×Z2 S (A.12)

In other words, Spin(4)C is the collection of equivalence classes [g, z] ∈ Spin(4)C with the equivalence relationship (g, z) = (−g, −z). We want to list a couple of homomorphisms.[14]

90 1.) λ : Spin(4)C → SO(4) given by λ([g, z]) = λ(g) (A.13)

2.) i : Spin(4) → Spin(4)C inclusion map (A.14)

3.) j : S1 → Spin(4)C inclusion map (A.15)

4.) l : Spin(4)C → S1 given by l([g, z]) = z2 (A.16)

5.) p : Spin(4)C → SO(4) × S1 given by p([g, z]) = (λ(g), z2) i.e. p = λ × l (A.17)

Minkowski space is a four dimensional vector space. The dimension being an even number means that the space of Dirac spinors, S decomposes into the direct sum of the so-called positive and negative Weyl spinors.[14] This is denoted

S = S+ ⊕ S− (A.18)

In this case, Cliﬀord multiplication by a non-zero vector v is a bijection, S± 7→ S∓.

In the space of Dirac spinors there exists a positive deﬁnite Hermitian inner product such that

(c(v).ψ, φ) = −(ψ, c(v).φ) (A.19)

for v ∈ M 4 and ψ, φ ∈ S. The spin representation κ : CL(M 4) → End(S) when restricted to the group Spin(4) ⊂ Spin(4)C ⊂ CL(M 4) is unitary with respect to this inner product. Furthermore,

det(κ(g)) = 1 (A.20)

for every g ∈ Spin(4). Therefore, the spin representation of the group Spin(4) is the group SU(S).[14] For example, if we have the metric signature (−, +, +, +) then the

91 spin representation of Spin(4) = Spin3,1 is given by the group SL(2, C) and has a representation space of Weyl spinors, i.e two component spinors S±. In the general case of [g, z] ∈ Spin(4)C we have the representation

κ([g, z])ψ = z.κ(g)ψ (A.21)

with determinant given by

det(κ([g, z])) = zdim(S) = z4 (A.22)

A.2 Spin Manifold and the Diﬀerential Structure of the Dirac Bundle

We want to extend the structure described above to the setting of manifolds and ﬁber bundles. A spin manifold is an oriented Riemannian manifold with a spin structure on its tangent bundle. An oriented Riemannian manifold M admits a spin structure (and is a spin manifold) if and only if its second Stiefel-Whitney class is zero.[13]

Let (M, g) be a 4 dimensional spin manifold. Let F denote the bundle of positively oriented orthonormal frames of the tangent bundle TM. F is a principal bundle with structure group SO(4). Suppose we have a Spin(4)C structure on TM. Since we have a Spin(4)C structure, we know that there exists an S1 = SO(2)-principal bundle

C C P1 such that the ﬁber product F×eP1 has a Spin(4) structure.[14] Our Spin(4) structure is given by the pair (W, Λ) where W is a Spin(4)C principal bundle and Λ is a double covering map given by

Λ: W → F×eP1 (A.23)

92 Recall from equation (A.17) that we also have the covering map p : Spin(4)C → SO(4) × S1. The SO(4) principal bundle allows us to deﬁne an associated four dimensional vector bundle.

4 4 T = F ×SO(4) M = W × λM (A.24)

T is isomorphic to the tangent bundle TM. The ﬁbers of T are Minkowski. As described above, we can view T as an associated vector bundle to the Spin(4)C principal bundle. Recall our spin representation κ. We can deﬁne another associated vector bundle to the Spin(4)C principal bundle by

∆ = W ×SpinC(4) S (A.25)

The ﬁbers of this associated vector bundle are a four complex dimensional space with a Hermitian metric. These ﬁbers are precisely the space of Dirac spinors. For this reason, we call ∆ the Dirac spinor bundle.

Finally, recall that each orthogonal transformation of M 4 induces and orthogonal transformation of CL(4). Hence we get a representation of SO(4) denoted

ρ : SO(4) → Aut(CL(4)) (A.26)

This allows us to deﬁne an associated bundle of Cliﬀord algebras.[13]

C(T ) = F ×ρ CL(4) (A.27)

In this way, at each point p ∈ M there is a complexiﬁed Cliﬀord algebra that is

generated by vectors in T |p ≡ TpM. This allows us the use of Clifford multiplication fiber-wise since we can view the tangent space as a fiber of the vector bundle associated

93 with the Spin(4)C principal bundle.

The Lie algebra of SO(4) we denote by so(4). The Lie algebra of S1 is iR and so the Lie algebra of the product SO(4) × S1 is so(4) ⊕ iR. The Lie algebra of C 3 Spin(4) we denote by spinC(4). Let x ∈ M and {ei}i=0 denotes the standard basis 4 of T |x = M . Then spinC(4) = m2 ⊕ iR where

m2 = span{c(ei).c(ej)|1 ≤ i < j ≤ 4} (A.28) m is a Lie algebra equipped with the commutator bracket which coincides with Cliﬀord multiplication

[x, y] = x.y − y.x (A.29)

We take the connection on the SO(4) principal bundle to be the Levi-Civita connection, which is the unique torsion-free metric connection. This is an so(4)- valued 1-form

Z : T F → so(4) (A.30)

1 We also ﬁx a connection on the S principal bundle P1. It is given by

A : TP1 → iR (A.31)

We use the product of these connections to deﬁne a connection in the ﬁber product

F×eP1.

Z × A : T (F×eP1) → so(4) ⊕ iR (A.32)

The map dΛ: T (W) → T (F×eP1) and the diﬀerential p∗ : m2 ⊕ iR → so(4) ⊕ iR allow us to lift the connection Z × A to a connection

94 Z^× A : T (W) → m2 ⊕ iR (A.33) which completes a commuting diagram. The connection Z^× A determines a covariant derivative in the Dirac bundle.[14]

Let X,Y be vector fields defined on M and let ψ be a spinor field. Then for the spinor covariant derivative with respect to any connection A in (A.31) we have

A A ∇Y (c(X).ψ) = c(X).(∇Y .ψ) + c(∇Y X).ψ (A.34) where ∇ is the Levi-Civita covariant derivative. [14] Also, the spinor covariant derivative is metric with respect to the Hermitian inner product on the Dirac bundle.

In other words, if X is a vector field and ψ, ψ1 are both spinor fields then the vector field acts as a derivation on the Hermitian inner product, i.e.

A A X(ψ, ψ1) = (∇X ψ, ψ1) + (ψ, ∇X ψ1) (A.35)

Finally, we want to give local expressions for the covariant derivative and curvature operators. Let e : U ⊂ M → F be a local section of the frame bundle given on an open set U. This is a positively oriented orthonormal frame ﬁeld. Let {Eij = ei ∧ ej} denote the standard basis of so(4), with matrix representation given by a 1 in the ith column, jth row and a −1 in the jth column, ith row. The local connection form of Z with respect to e we denote Ze. It is given by

e X Z = ωijEij (A.36) i

where ωij = g(∇ei , ej) are the standard 1-forms for the Levi-Civita connection. Let 1 s : U ⊂ M → P1 be a ﬁxed section of the S principal bundle. Then the local connection form of A with respect to s we denote As. It is given by a map

95 s A : TU → iR (A.37)

Then we have the local connection form of the product Z × A with respect to the

product of sections e × s : U → F×eP1. It is given by

e×s X s Z × A = ( ωijEij,A ) (A.38) i

Let the lift of the product of sections e × s be given by e]× s : U → W. It is a fact

[13] that the Lie algebras spin(4) = m2 and so(4) are isomorphic, with isomorphism given by

1 E 7→ c(e ).c(e ) (A.39) ij 2 i j With (A.38) and (A.39) we have the local connection form of Z^× A with respect ]e×s to e]× s. It is denoted Z^× A and given by

]e×s 1 1 Z^× A = ( ω c(e ).c(e ), As) (A.40) 2 ij i j 2 (A.40) allows us to give a local expression of the covariant derivative on the Dirac bundle. It is the following.[14]

1 X 1 ∇Aψ = dψ + ω c(e ).c(e ).ψ + Asψ (A.41) 2 ij i j 2 i

ΩZ : T F × F → so(4) (A.42)

given by

96 Z X Ω = ΩijEij (A.43) i

P3 where Ωij = dωij + k=0 ωik ∧ ωkj. The curvature 2-form for the connection A is given by

ΩA = dA (A.44)

We are then able to lift the product to a curvature 2-form for the connection Z^× A. It is given by

1 X 1 ΩZ^×A = Λ∗(Ω )c(e ).c(e ) ⊕ Λ∗(dA) (A.45) 2 ij i j 2 i

1 X 1 ∇A(∇Aψ) = Ω c(e ).c(e ).ψ + dA.ψ (A.47) 2 ij i j 2 i

A.3 Dirac Operator and the Schr¨odinger-Lichnerowicz formula

Let ∇A denote the covariant derivative on the Dirac bundle. Recall (A.34) and (A.35). These equations show how the covariant derivative ∇A interacts with the Hermitian

97 inner product (, ) defined on the Dirac bundle. Let {ei} = e be the positively oriented orthonormal frame field given above. We first define the Laplace operator on the Dirac bundle. If ψ is a Dirac spinor field, then the Laplace operator on the Dirac bundle, which we denote ∆A(ψ), is given by

X X ∆A(ψ) = − ∇A ∇A ψ − div(e )∇A ψ (A.49) ei ei i ei i i where the divergence is given by

X − div(ej) = g(∇ek ej, ek)ek (A.50) k

Next we define the Dirac operator. The Dirac operator, denoted D , is a differential operator on the space of spinors which is defined as the contraction of covariant differentiation and Clifford multiplication.[14] Namely,

X Dψ = c(e ).∇A ψ (A.51) i ei i We now give the statement of the Schr¨odinger-Lichnerowicz Formula.

Proposition A.1[Schr¨odinger-Lichnerowicz Formula] Denote by R the scalar curvature of the Riemannian manifold and let dA = ΩA be the imaginary-valued curvature 2-form of the connection A in the S1-principal bundle associated with the Spin(4)C structure. Then one has for a Dirac spinor ﬁeld ψ,

2 A 1 1 D ψ = ∆ ψ + Rψ + dA.ψ 4 2

The goal now is to establish the Schr¨odinger-Lichnerowicz formula. The formula is a way to relate the square of the Dirac operator and the Laplace operator on the

Dirac bundle. Let P ∈ M. We have an orthonormal frame {ei} on TP M. Recall the Cliﬀord relationships from (A.1) and (A.2). In particular, at P we have

98 c(ei).c(ei) = −1 (A.52) c(ei).c(ej) + c(ej).c(ei) = −2δij (A.53)

99 We now compute the diﬀerence between the square of the Dirac operator and the Spin connection Laplacian.

X X X X D 2 − ∆A = c(e ).∇A ( c(e ).∇A ) + ∇A ∇A + div(e )∇A (A.54) i ei j ej ei ei i ei i j i i X X X = {c(e ).c(∇ e ).∇A + c(e ).c(e ).∇A ∇A } + ∇A ∇A + div(e )∇A i ei j ej i j ei ej ei ei i ei i,j i i (A.55) X X X = g(∇ e , e )c(e ).c(e ).∇A + c(e ).c(e ).∇A ∇A + ∇A ∇A ei j k i k ej i j ei ej ei ei i,j,k i,j i (A.56) X + div(e )∇A i ei i X X = g(∇ e , e )c(e ).c(e ).∇A + g(∇ e , e )c(e ).c(e ).∇A + ei j k i k ej ei j k i k ej j,i6=k j,i=k (A.57) X X X X c(e ).c(e ).∇A ∇A + c(e ).c(e ).∇A ∇A + ∇A ∇A + div(e )∇A i j ei ej i j ei ej ei ei i ei i6=j i=j i i X X X = g(∇ e , e )c(e ).c(e ).∇A − div(e )∇A + c(e ).c(e ).∇A ∇A − ei j k i k ej i ei i j ei ej j,i6=k i i6=j (A.58) X X X ∇A ∇A + ∇A ∇A + div(e )∇A ei ei ei ei i ei i i i X X = g(∇ e , e )c(e ).c(e ).∇A + c(e ).c(e ).∇A ∇A (A.59) ei j k i k ej i j ei ej j,i6=k i6=j X X = − g(e , ∇ e )c(e ).c(e ).∇A + c(e ).c(e ).∇A ∇A (A.60) j ei k i k ej i j ei ej j,i6=k i6=j X X = − g(e , ∇ e − ∇ e )c(e ).c(e ).∇A + c(e ).c(e ).[∇A ∇A − ∇A ∇A ] j ei k ek i i k ej i j ei ej ej ei j,i

100 X X = g(e , [e , e ])c(e ).c(e ).∇A + c(e ).c(e ).[∇A ∇A − ∇A ∇A ] (A.62) j k i i k ej i j ei ej ej ei j,i

Recall equation (A.48) and that

A A A (∇ ∇ ψ)(ei, ej) = R (ei, ej)ψ (A.68)

This gives us for (A.67)

1 X 1 1 1 X 1 X c(e ).[− Ric(e ) + (e dA)] = − c(e )Ric(e ) + c(e )(e dA) 2 i 2 i 2 iy 4 i i 4 i iy i i i (A.69) 1 1 = R + dA (A.70) 4 2

Therefore,

2 A 1 1 D = ∆ + R + dA (A.71) 4 2

101 A.4 Witten’s expression for the mass

The above formula in equation (A.71) is key to the derivation of Witten’s expression for mass. Suppose that M is an asymptotically ﬂat,complete, n-dimensional, spin manifold with metric

gij = δij + hij (A.72)

2,α n−2 1 with hij ∈ W−τ (M) for τ > 2 and R ∈ L . Then Witten’s expression for the mass in n-dimensions is given by

2 2 c(n)mass(g) = (4k∇e ψk + Rkψk )dVg (A.73) ˆM We are considering a general spin structure here instead of the more complicated complexification. This means we do not specifically have a “Dirac” spinor, i.e. 4 component spinor. In this case, the Schrödinger-Lichnerowicz Formula takes the following form when applied to a spinor ψ.[19]

2 ∗ 1 D ψ = ∇ ∇ψ + Rψ (A.74) 4

3 In an orthonormal frame {ei}i=0, we recall the isomorphism between so(4) and spin(4) given by

1 e ∧ e 7→ c(e ).c(e ) (A.75) i j 2 i j This allows us to lift the Levi-Civita connection Ze to a connection on an associated spin bundle. The connection on the spin bundle is

e 1 Zee = ωijc(ei).c(ej) (A.76) 2

102 where ωij are the Levi-Civita connection 1-forms. This gives us the expression for the covariant derivative of spinors in a local frame. We denote the covariant derivative that acts on spinors as ∇e . The formula for the covariant derivative is given by

1 X ∇e ψ = dψ + ωijc(ei).c(ej).ψ (A.77) 2 i

Now, let ψ be a spinor that is a solution to Dirac’s equation, i.e.

Dψ = 0 (A.80)

2 that is asymptotic to a given constant spinor normalized by |ψ0| → 1. The required 2,α spinor ψ = ψ0 + ψ1, where ψ1 ∈ W−τ , exists.[17]

We need to prove an identity. Let (, ) denote the positive deﬁnite inner product on the bundle of spinors, i.e. from (A.19). Recall that the Laplacian and Dirac operator

1 i j are both self-adjoint. Let σij = 2 [c(e ), c(e )] so that

1 i j σij.∇fj = [c(e ), c(e )].∇fj (A.81) 2 ij i j = (δ + c(e ).c(e )).∇fj (A.82)

i = ∇fi + e . D (A.83)

Then,

103 i d{(ψ, σij.∇e jψ)eiydV olg} = d((ψ, ∇e iψ)eiydV olg) + d((ψ, e . Dψ )eiydV olg} (A.84) i = {(∇e iψ, ∇e iψ) + (ψ, ∇e i∇e iψ) + (∇e iψ, e . Dψ )+ (A.85)

i 2 (ψ, c(∇ie ). Dψ ) + (ψ, D ψ)}dV olg

2 i = {k∇e ψk + (ψ, ∇e i∇e iψ) − (e .∇e iψ, Dψ )+ (A.86)

i 2 (ψ, c(∇ie ). Dψ ) + (ψ, D ψ)}dV olg

2 = {k∇e ψk + (ψ, ∇e i∇e iψ) − ( Dψ, Dψ )+ (A.87)

i 2 (ψ, c(∇ie ). Dψ ) + (ψ, D ψ)}dV olg

2 2 = {k∇e ψk + (ψ, ∇e i∇e iψ) − k Dψ k + (A.88)

i 2 (ψ, c(∇ie ). Dψ ) + (ψ, D ψ)}dV olg

2 2 2 = {k∇e ψk − (ψ, −∇e i∇e iψ) − k Dψ k + (ψ, D ψ)+ (A.89)

1 1 i (ψ, Rψ) − (ψ, Rψ) + (ψ, c(∇ie ). Dψ )}dV olg 4 4 1 1 = {k∇e ψk2 − (ψ, ∇∗∇ψ) + (ψ, Rψ) − (ψ, Rψ)− (A.90) 4 4 2 2 k Dψ k + (ψ, D ψ)}dV olg

2 R 2 2 = {k∇e ψk + kψk − k Dψ k }dV olg, (A.91) 4 using (A.74). This is the identity we wanted to prove. We state it here for clarity.

2 R 2 2 {k∇e ψk + kψk − k Dψ k }dV olg = d{(ψ, σij.∇e jψ)ei dV olg} (A.92) 4 y This is the key to Witten’s formula for mass. We use this identity for our spinor described above, i.e. ψ = ψ0 + ψ1. We integrate this over M and identify the boundary term on the right with the ADM mass. Recall

c(n)mass(g) = lim (∂jgij − ∂igjj)ei dV olg (A.93) R→∞ y ˆ∂BR

104 Our next step is to integrate (A.91) with ψ = ψ0 + ψ1 over the region MR = {r ≤ R} and use Stokes Theorem. Note that the left hand side is real, and so we are only concerned with the real part of the right hand side.

2 R 2 2 k∇e ψk + kψk − k Dψ k }dV olg = d{(ψ, σij.∇e jψ)eiydV olg} (A.94) ˆMR 4 ˆMR

= (ψ, σij.∇e jψ)eiydV olg (A.95) ˆ∂MR

= (ψ0 + ψ1, σij.∇e jψ0 + σij.∇e jψ1)eiydV olg ˆ∂MR (A.96)

We now reference an identity given by Bartnik in [17]. It is the following relationship

d{(φ, σij.χ)(ei ∧ ej)ydV olg} = {(φ, σij.∇e jχ) − (σij.∇e jφ, χ)}eiydV olg (A.97)

The identity in (A.97) give us an equivalent expression for (A.96).

(ψ0, σij.∇e jψ0) + d{(ψ0, σij.ψ1)(ei ∧ ej)ydV olg} + (ψ1, σij.∇e jψ)+ (A.98) ˆ∂MR

(σij.∇e jψ0, ψ1)eiydV olg

In (A.98) we note that the last two terms will not contribute to the integral at inﬁnity

2 because of the falloﬀ conditions on ψ1. The second term disappears because d = 0.

This leaves us, at inﬁnity, with the term (ψ0, σij.∇e iψ0). Evaluating this term using

(A.78) and recalling that ψ0 is constant at inﬁnity gives us

105 1 X (ψ0, σij.∇e jψ0) = (ψ0, σijdψ0 − ωkl(ej)σijc(ek).c(el).ψ0) (A.99) 4 k,l 1 X = (ψ , − ω (e )σ .c(e ).c(e ).ψ ) (A.100) 0 4 k,l j ij k l 0 k,l 1 X = (ψ , − ω (e )σ .σ .ψ ) (A.101) 0 4 k,l j ij kl 0 i,j,k,l 1 X = − ω (e )(ψ , σ .σ .ψ ) (A.102) 4 k,l j 0 ij kl 0 i,j,k,l

Let σijkl = c(ei).c(ej).c(ek).c(el) if i 6= j 6= k 6= l and 0 otherwise. We can easily

verify the fact that σij is skew hermitian. Also, looking at the left hand side of (A.94) we see that we are only interested in the real part of the right hand side. Therefore, equation (A.102) simpliﬁes to the following.[17]

1 1 ω (e )kψ k2 − ω (e )(ψ , σ .ψ ) (A.103) 2 ij j 0 4 kl j 0 ijkl 0

For the second term note that σijkl is antisymmetric. This term turns out to be equal to the divergence plus terms on the order or r−2τ−1. Therefore it will not contribute to the boundary integral at inﬁnity. j Now we introduce coordinates that were used by Bartnik.[17]. Let ei = ei ∂j be an orthonormal frame near inﬁnity satisfying:

j 2,α ei − δij ∈ W−τ (MR) (A.104)

This frame is called asymptotically constant. For the ﬁrst term in (A.103) we want to write out ωij in terms of Christoﬀel symbols and the frame.

i −2τ−1 ωij(ej) = Γijj + ∂j(ej) + O(r ) (A.105)

106 i Next we want to decompose the frame as e = (ej) = δ + s + a where s is a symmetric part and a is the antisymmetric part. This gives us the following expression.[17]

1 ω (e ) = (∂ g − ∂ g ) + ∂ a + O(r−2τ−1) (A.106) ij j 2 j ij i jj j ji Finally note that

i j −2τ−1 ∂jajieiydV olg = d(aij(dx ∧ dx )ydV olg) + O(R ) (A.107)

Therefore, taking R → ∞ in (A.94)-(A.96), recalling (A.103), (A.106)-(A.107) and the fact that our spinor ψ is in the kernel of the Dirac operator gives us

2 R 2 1 c(n) k∇e ψk + kψk dV olg = lim (∂jgij − ∂igjj)ei dV olg = mass(g) R→∞ y ˆM 4 4 ˆ∂MR 4 (A.108) We note that under the decay conditions of asymptotic ﬂatness, this integral converges.

107 Vita

Joshua Michael Lipsmeyer was born on October 29, 1983 in Bigelow, Arkansas. He was raised in a small town by loving parents Taneau and Gary Lipsmeyer. He was the oldest of four children, with siblings Brandi Edwards, Joey Lipsmeyer, and Shannon Howell. Josh received his high school diploma from Bigelow High School in 2002 with honors. After a successful career in industry, Josh went on to pursue undergraduate studies at the University of Arkansas at Little Rock where he graduated Magna Cum Laude with a Bachelor of Science degree in mathematics with a minor in physics. In 2012, Josh received a graduate teaching assistantship from the University of Tennessee. There, his area of study was geometric analysis and general relativity under the guidance of Dr. Alex Freire. In August 2015 he was awarded his Masters of Science in mathematics.

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